Jtii,,AHUUii-y 


Ex  Xi^rli- 
<3xank  iPid^r  ^icauL 


Digitized  by  tine  Internet  Arciiive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


littp://www.archive.org/details/foundationsofmatOOcaru 


THE   FOUNDATIONS  OF 
MATHEMATICS 


A  CONTRIBUTION   TO 


THE  PHILOSOPHY  OF  GEOMETRY 


BY 


DR.   PAUL  CARUS 


o    d(.<t<;    ati    yciofj.f.TpeL.  —  I'LATO. 


CHICAGO 
Tin-:  ( )I'EN  COURT  rUBLISHING  CO. 

LONDON    AGENTS 
KEGAN   PAUL,  TRENCH,  TrCbNER  &  CO.,  LTD. 


COPYRIGHT  BY 

THE  OPEN  COURT  PUB.  CO. 
igo8 


URL 


TABLE  OF  CONTEXTS. 

THE    SEARCH    FOR    THE    FOUNDATIONS    OF   GEOM- 
ETRY:   HISTORICAL   SKETCH. 

PACE 

Axioms  and  the  Axiom  of  Parallels i 

Metageometry 5 

Precursors 7 

Gauss II 

Riemann 15 

Lobatchevsky 20 

Bolyai 22 

Later  Geometricians 24 

Grassmann 27 

Euclid  Still  Unimpaired 31 

THE   PHILOSOPHICAL   BASIS  OF  MATHEMATICS. 

The  Philosophical  Problem 35 

Transcendentalism  and   Empiricism 38 

The  A  Priori  and  the  Purely  Ff)rmal 40 

Anyness  and  its  Universality 46 

Apriority  of  Different  Degrees 49 

Space  as  a  Spread  of  Motion 56 

Uniqueness  of  Pure  Space 6r 

Mathematical  Space  and  Physiological  Space 63 

Homogeneity  of  Space  Due  to  Ab.straction 66 

Even  Boundaries  as  Standards  of  Measurement 6g 

The  Straight  Line  Indispensal)le 72 

The  Superrcal 76 

Discrete  Units  and  the  Continuum 78 

MATHEMATICS   AND   METAGEOMK  TRY. 

Different  Geometrical  Systems 82 

Tridimensionality 84 

Three  a  Concept  of  {'.ounfiary 88 


IV  THE  FOUNDATIONS  OF  MATHEMATICS. 

PAGE 

Space  of  Four  Dimensions 90 

The  Apparent  A.rbitrariness  of  the  A  Priori 96 

Definiteness  of  Construction 99 

One  Space,  But  Various  Systems  of  Space  Measurement..  .  104 
Fictitious  Spaces  and  the  Apriority  of  All  Space  Measure- 
ment   109 

Infinitude 116 

Geometry  Remains  A  Priori 119 

Sense-Experience  and  Space 122 

The  Teaching  of  Mathematics 127 

EPILOGUE 132 

INDEX 139 


THE   SEARCH   FOR   THE   FOUXDATIOXS 
OF  GEOxAIETRY:  HISTORICAL  SKETCH. 

AXIOMS   AND   THE    AXIOM    OF    PARALLELS. 

MATHEMATICS  as  commonly  taught  in  our 
schools  is  based  upon  axioms.  These  axioms 
so  called  are  a  few  simple  formulas  which  the  be- 
ginner must  take  on  trust. 

Axioms  are  defined  to  be  self-evident  ])ropo- 
sitions,  and  are  claimed  to  be  neither  demonstrable 
nor  in  need  of  demonstration.  They  are  statements 
which  are  said  to  command  the  assent  of  every 
one  who  comprehends  their  meaning. 

The  word  axiom'  means  "honor,  reputation, 
high  rank,  authority,"  and  is  used  by  Aristotle, 
almost  in  the  modern  sense  of  the  term,  as  "a  self- 
evident  highest  principle,"  or  "a  truth  so  obvious 
as  to  be  in  no  need  of  proof."  It  is  derived  from 
the  \'erl)  a^covu ,  "lo  deem  worthw  to  think  fit,  to 
maintain,"  and  is  cognate  with  a^tos,  '"worth"  or 
"worthy." 

luiclid  does  not  use  the  term  "axiom."  lie 
starts  with  Defim'tions."  which  describe  the  mean- 
ings of  point,  line,  surface,  plane,  angle,  etc.     lie 


2  THE   FOUNDATIONS   OF   MATHEMATICS. 

then  proposes  Postulates"  in  which  he  takes  for 
granted  that  we  can  draw  straight  Hnes  from  any 
point  to  any  other  point,  and  that  we  can  prolong 
any  straight  line  in  a  straight  direction.  Finally, 
he  adds  what  he  calls  Common  Notions'*  which  em- 
body some  general  principles  of  logic  (of  pure  rea- 
son) specially  needed  in  geometry,  such  as  that 
things  which  are  equal  to  the  same  thing  are  equal 
to  one  another;  that  if  equals  be  added  to  ec[uals, 
the  wholes  are  equal,  etc. 

I  need  not  mention  here  perhaps,  since  it  is  a 
fact  of  no  consequence,  that  the  readings  of  the 
several  manuscripts  vary,  and  that  some  proposi- 
tions (e.  g.,  that  all  right  angles  are  equal  to  one 
another)  are  now  missing,  now  counted  among  the 
postulates,  and  now  adduced  as  common  notions. 

The  commentators  of  Euclid  who  did  not  under- 
stand the  difference  between  Postulates  and  Com- 
mon Notions,  spoke  of  both  as  axioms,  and  even 
to-day  the  term  Common  Notion  is  mostly  so  trans- 
lated. 

In  our  modern  editions  of  Euclid  we  find  a 
statement  concerning  parallel  lines  added  to  either 
the  Postulates  or  Common  Notions.  Originally  it 
appeared  in  Proposition  29  where  it  is  needed  to 
prop  up  the  argument  that  would  prove  the  equality 
of  alternate  angles  in  case  a  third  straight  line  falls 
upon  parallel  straight  lines.  It  is  there  enunciated 
as  follows: 

"But  those  straight  Hnes  which,  with  another  straight 

'  aiTirj/jLara.  ^  Koivat  ivvoiai. 


HISTORICAL  SKETCH.  3 

line  fallin<T^  upon  them,  make  the  interior  angles  on  the  same 
side  less  than  two  right  angles,  do  meet  if  continually  pro- 
duced." 

Now  this  is  exactly  a  point  that  calls  for  proof. 
Proof  was  then,  as  ever  since  it  has  remained,  alto- 
gether lacking.  So  the  proposition  was  formulated 
dogmatically  thus: 

"If  a  straight  line  meet  two  straight  lines,  so  as  to  make 
the  two  interior  angles  on  the  same  side  of  it  taken  together 
less  than  two  right  angles,  these  straight  lines  being  con- 
tinually produced,  shall  at  length  meet  upon  that  side  on 
which  are  the  angles  which  are  less  than  two  right  angles." 

And  this  proposition  has  heen  transferred  by 
the  editors  of  Euclid  to  the  introductory  portion  of 
the  book  where  it  now  appears  either  as  the  fifth 
Postulate  or  the  eleventh,  twelfth,  or  thirteenth 
Common  Notion.  The  latter  is  obviously  the  less 
appropriate  place,  for  tiie  idea  of  ])arallelism  is 
assuredly  not  a  Common  Notion ;  it  is  not  a  rule 
of  pure  reason  such  as  would  be  an  essential  con- 
dition of  all  thinking,  reasoning,  or  logical  argu- 
ment. And  if  we  do  not  give  it  a  place  of  its  own, 
it  should  cither  be  classed  among  the  postulates,  or 
recast  so  as  to  become  a  pure  definition.  Tt  is  usu- 
ally referred  to  as  "the  axiom  of  ])arallels." 

Tt  seems  lo  me  that  no  one  can  read  the  axiom  of 
parrdlels  as  it  stands  in  F.urlid  without  receiving 
tlie  imprcssirm  that  tlie  statement  was  affixed  bv  a 
later  redactor.  Evvu  in  I 'rcjpositirm  29,  the  original 
place  of  its  insertion,  ii  comes  in  as  an  afterthought  ; 
and  if   I'jiclid  himself  had  considered  the  difficult v 


4  THE   FOUNDATIONS   OF   MATHEMATICS. 

of  the  parallel  axiom,  so  called,  he  would  have  placed 
it  among  the  postulates  in  the  first  edition  of  his 
book,  or  formulated  it  as  a  definition.^ 

Though  the  axiom  of  parallels  must  be  an  inter- 
polation, it  is  of  classical  origin,  for  it  was  known 
even  to  Proclus  (410-485  A.  D.),  the  oldest  com- 
mentator of  Euclid. 

By  an.  irony  of  fate,  the  doctrine  of  the  parallel 
axiom  has  become  more  closely  associated  with 
Euclid's  name  than  anything  he  has  actually  writ- 
ten, and  when  we  now  speak  of  Euclidean  geometry 
we  mean  a  system  based  upon  that  determination  of 
parallelism. 

We  may  state  here  at  once  that  all  the  attempts 
made  to  derive  the  axiom  of  parallels  from  pure 
reason  were  necessarily  futile,  for  no  one  can  prove 
the  absolute  straightness  of  lines,  or  the  evenness  of 
space,  by  logical  argument.  Therefore  these  con- 
cepts, including  the  theory  concerning  parallels, 
cannot  be  derived  from  pure  reason ;  they  are  not 
Common  Notions  and  possess  a  character  of  their 
own.  But  the  statement  seemed  thus  to  hang  in  the 
air.  and  there  appeared  the  possibility  of  a  geom- 
etry, and  even  of  several  geometries,  in  whose  do- 
mains the  parallel  axiom  would  not  hold  good.  This 
large  field  has  been  called  metageometry,  hyper- 

^  For  Professor  Halsted's  ingenious  interpretation  of  the  origin 
of  the  parallel  theorem  see  The  Monist,  Vol.  IV.  No.  4,  p.  487.  He 
believes  that  Euclid  anticipated  metageometry,  but  it  is  not  probable 
that  the  man  who  wrote  the  argument  in  Proposition  29  had  the  fifth 
Postulate  before  him.  He  would  have  referred  to  it  or  stated  it  at 
least  approximatel}'  in  the  same  words.  But  the  argument  in  Propo- 
sition 29  differs  considerably  from  the  parallel  axiom  itself. 


HISTORICAL  SKETCH.  5 

geometry,  or  pangeometry,  and  may  be  regarded 
as  due  to  a  generalization  of  the  space-conception 
involving  what  might  be  called  a  metaphysics  of 
mathematics. 

METAGEOMETRY. 

Mathematics  is  a  most  conservative  science.  Its 
system  is  so  rigid  and  all  the  details  of  geometrical 
demonstration  are  so  complete,  that  the  science  was 
comrnonly  regarded  as  a  model  of  perfection.  Thus 
the  philosophy  of  mathematics  remained  undevel- 
oped almost  two  thousand  years.  Not  that  there 
were  not  great  mathematicians,  giants  of  thought, 
men  like  the  Bernoullis.  Leibnitz  and  Newton,  Euler, 
and  others,  worthy  to  be  named  in  one  breath  with 
Archimedes,  Pythagoras  and  iCuclid,  but  they  ab- 
stained from  entering  into  philosophical  specula- 
tions, and  the  very  idea  of  a  ]iangeometrv  remained 
foreign  to  them.  They  ma}-  priwUely  have  reflected 
on  the  su1)ject,  but  they  did  not  give  utterance  to 
their  thoughts,  at  least  the}'  left  no  records  of  them 
to  posterity. 

It  would  l)e  wrong,  however,  to  assume  tliat  the 
mathematicians  of  former  ages  were  not  conscious 
of  the  difficulty.  11iey  always  felt  that  there  was 
a  flaw  in  the  luiclidean  foundation  of  geometry, 
but  they  were  satisfied  to  supply  an\'-  need  of  basic 
principles  in  the  shai)e  of  axioms,  and  il  lias  become 
(|uile  customary  (  1  might  almost  say  orthodox)  to 
say  that  mathematics  is  based  upon  axioms.  In  fact, 
peoi)le  enjoyed  the  idea  that  mathematics,  the  most 


6  THE   FOUNDATIONS   OF   MATHEMATICS. 

lucid  of  all  the  sciences,  was  at  bottom  as  mysterious 
as  the  most  mystical  dogmas  of  religious  faith. 

Metageometry  has  occupied  a  peculiar  position 
among  mathematicians  as  well  as  with  the  public  at 
large.  The  mystic  hailed  the  idea  of  "?z-dimensional 
spaces,"  of  "space  curvature"  and  of  other  concep- 
tions of  which  we  can  form  expressions  in  abstract 
terms  but  which  elude  all  our  attempts  to  render 
them  concretely  present  to  our  intelligence.  He 
relished  the  idea  that  by  such  conceptions  mathe- 
matics gave  promise  to  justify  all  his  speculations 
and  to  give  ample  room  for  a  multitude  of  notions 
that  otherwise  would  be  doomed  to  irrationality. 
In  a  word,  metageometry  has  always  proved  attrac- 
tive to  erratic  minds.  Among  the  professional  math- 
ematicians, however,  those  who  were  averse  to  phil- 
osophical speculation  looked  upon  it  with  deep  dis- 
trust, and  therefore  either  avoided  it  altogether  or 
rewarded  its  labors  wath  bitter  sarcasm.  Prominent 
mathematicians  did  not  care  to  risk  their  reputation, 
and  consequently  many  valuable  thoughts  remained 
unpublished.  Even  Gauss  did  not  care  to  speak 
out  boldly,  but  communicated  his  thoughts  to  his 
most  intimate  friends  under  the  seal  of  secrecy,  not 
unlike  a  religious  teacher  who  fears  the  odor  of 
heresy.  He  did  not  mean  to  suppress  his  thoughts, 
but  he  did  not  want  to  bring  them  before  the  public 
unless  in  mature  shape.  A  letter  to  Taurinus  con- 
cludes with  the  remark: 

"Of  a  man  who  has  proved  himself  a  thinking  mathe- 
matician, T  fear  not  that  he  will  misunderstand  what  I  say, 


HISTORICAL  SKETCH.  7 

but  under  all  circumstances  you  have  to  regard  it  merely  as 
a  private  communication  of  which  in  no  wise  public  use,  or 
one  that  may  lead  to  it,  is  to  be  made.  Perhaps  I  shall  pub- 
lish them  myself  in  the  future  if  I  should  gain  more  leisure 
than  m)'  circumstances  at  present  permit. 

"C.  F.  Gauss. 
"GoETTiNGEx,  8.  November,  1824." 

But  Gauss  never  did  publish  anything  upon  this 
topic  ahhough  the  seeds  of  his  thought  thereupon 
fell  upon  fertile  ground  and  bore  rich  fruit  in  the 
works  of  his  disciples,  foremost  in  those  of  Riemann. 

PRECURSORS. 

The  first  attempt  at  improvement  in  the  matter 
of  parallelism  was  made  by  Nasir  Eddin  (  1201- 
1274)  whose  work  on  Euclid  was  printed  in  Arabic 
in  1594  in  Rome.  His  labors  were  noticed  by  John 
W'allis  who  in  165 1  in  a  Latin  translation  com- 
municated Nasir  Eddin's  exposition  of  the  fifth  Pos- 
tulate to  the  mathematicians  of  the  University  of 
Oxford,  and  then  propounded  his  own  views  in  a 
lecture  delivered  on  July  11,  1663.  Nasir  Eddin 
takes  his  stand  upon  the  postulate  that  two  straight 
h'nes  which  cut  a  third  straight  line,  the  one  at 
right  angles,  the  other  at  some  other  angle,  will 
converge  on  the  side  where  the  angle  is  acute  and 
diverge  where  it  is  obtuse.  W'nllis,  in  liis  endeavor 
to  prove  this  pnstnlnic,  starts  witli  tlie  auxiliary 
theorem  : 

"If  a   limited   straight   lino   which   lies  upon   an   un- 
limited straight  line  be  prolonged  in  a  straight  direction, 


5  THE   FOUNDATIONS   OF   MATHEMATICS. 

its   prolongation   will    fall   ui)on   the   unlimited   straight 
line." 

There  is  no  need  of  entering  into  the  details  of 
his  proof  of  this  auxiliary  theorem.  We  may  call 
his  theorem  the  proposition  of  the  straight  line  and 
may  grant  to  him  that  he  proves  the  straightness  of 
the  straight  line.  In  his  further  argument  Wallis 
shows  the  close  connection  of  the  problem  of  paral- 
lels with  the  notion  of  similitude. 

Girolamo  Saccheri,  a  learned  Jesuit  of  the  seven- 
teenth century,  attacked  the  problem  in  a  new  way. 
Saccheri  was  born  September  5,  1667,  at  San  Remo. 
Having  received  a  good  education,  he  became  a 
member  of  the  Jesuit  order  March  24,  1685,  and 
served  as  a  teacher  of  grammar  at  the  Jesuit  College 
di  Brera,  in  Milan,  his  mathematical  colleague  be- 
ing Tommaso  Ceva  (a  brother  of  the  more  famous 
Giovanni  Ceva).  Later  on  he  became  Professor  of 
Philosophy  and  Polemic  Theology  at  Turin  and  in 
1697  at  Pavia.  He  died  in  the  College  di  Brera 
October  25,  1733. 

Saccheri  saw  the  close  connection  of  parallelism 
with  the  right  angle,  and  in  his  work  on  Euclid^  he 
examines  three  possibilities.  Taking  a  quadrilateral 
ABCD  with  the  angles  at  A  and  B  right  angles 
and  the  sides  AC  and  BD  equal,  the  angles  at  C  and 
D  are  without  difficulty  shown  to  be  equal  each  to 
the  other.  They  are  moreover  right  angles  or  else 
they  are  either  obtuse  or  acute.     He  undertakes  to 

^  Euclidcs  ab  omni  nacvo  vindicattis:  sive  conatus  geomefn'ctts 
quo  stabiliuntur  prima  ipsa  nniversae  geometriae  principia.  Auctore 
Hieronymo  Saccherio  Societatis  Jesu.     Mediolani,  1773. 


HISTORICAL  SKETCH.  9 

prove  the  absurdity  of  these  two  latter  suppositions 
so  as  to  leave  as  the  only  solution  the  sole  possibility 
left,  viz.,  that  they  must  be  right  angles.  But  he 
finds  difficulty  in  pointing  out  the  contradiction  to 
which  these  assumptions  may  lead  and  thus  he  opens 
a  path  on  which  Lobatchevsky  (1793-1856)  and 
Bolyai  (1802-1860)  followed,  reaching  a  new  view 
which  makes  three  geometries  possible,  viz.,  the 
geometries  of  (i)  the  acute  angle,  (2)  the  obtuse 
angle,  and  (3)  the  right  angle,  the  latter  being  the 
Euclidean  geometry,  in  which  the  theorem  of  paral- 
lels holds. 


D 


While  Saccheri  seeks  the  solution  of  the  problem 
through  the  notion  of  the  right  angle,  the  German 
mathematician  Lambert  starts  from  the  notion  of 
the  angle-sum  of  the  triangle. 

Johann  Heinrich  Lambert  was  born  August  26, 
1728,  in  Miihlhausen,  a  city  which  at  that  time  was 
a  part  of  Switzerland.  Tie  died  in  1777.  His  TJic- 
ory  of  flic  Parallel  Lines,  written  in  1766,  was  not 
published  till  171^6,  nine  years  after  his  death,  by 
Ilcrnoulli  and  Ilindcnbnrg  in  liie  Magaziii  fiir  die 
reiiie  and  aiii^eiuandte  MatJiematik. 

Lambert  j)oints  out  that  there  arc  three  possi- 
bilities: the  sum  of  the  angles  of  a  triangle  may  be 


10  THE   FOUNDATIONS   OF   MATHEMATICS. 

exactly  equal  to,  more  than,  or  less  than  i8o  degrees. 
The  first  will  make  the  triangle  a  figure  in  a  plane, 
the  second  renders  it  spherical,  and  the  third  pro- 
duces a  geometry  on  the  surface  of  an  imaginary 
sphere.  As  to  the  last  hypothesis  Lambert  said  not 
without  humor  :'^ 

"This  result*  possesses  somethinq-  attractive  which  easily 
suggests  the  wish  that  the  third  hypothesis  might  be  true." 

He  then  adds:^ 

"But  I  do  not  wish  it  in  spite  of  these  advantages,  be- 
cause there  would  be  innumerable  other  inconveniences. 
The  trigonometrical  tables  would  become  infinitely  more 
complicated,  and  the  similitude  as  well  as  proportionality  of 
figures  would  cease  altogether.  No  figure  could  be  repre- 
sented except  in  its  own  absolute  size ;  and  astronomy  would 
be  in  a  bad  plight,  etc." 

Lobatchevsky's  geometry  is  an  elaboration  of 
Lambert's  third  hypothesis,  and  it  has  been  called 
"imaginary  geometry"  because  its  trigonometric 
formulas  are  those  of  the  spherical  triangle  if  its 
sides  are  imaginary,  or,  as  Wolfgang  Bolyai  has 
shown,  if  the  radius  of  the  sphere  is  assumed  to  be 
imaginary  =(V — i)r, 

France  has  contributed  least  to  the  literature  on 
the  subject.  Augustus  De  Morgan  records  the  fol- 
lowing story  concerning  the  efforts  of  her  greatest 
mathematician  to  solve  the  Euclidean  problem.   La- 

^  P.  3SI,  last  line  in  the  Magacin  fiir  die  rcinc  uud  angczvandtc 
Mathentatik,  1786. 

'  Lambert  refers  to  the  proposition  that  the  mooted  angle  might 
be  less  than  90  degrees. 

'Ibid.,  p.  352. 


HISTORICAL  SKETCH.  II 

grange,  he  says,  composed  at  the  close  of  his  hfe 
a  discourse  on  parallel  lines.  He  began  to  read  it 
in  the  Academy  but  suddenly  stopped  short  and 
said:  "II  faut  que  j'y  songe  encore."  With  these 
words  he  pocketed  his  papers  and  never  recurred 
to  the  subject. 

Legendre's  treatment  of  the  subject  appears  in 
the  third  edition  of  his  elements  of  Euclid,  but  he 
omitted  it  from  later  editions  as  too  difficult  for  be- 
ginners. Like  Lambert  he  takes  his  stand  upon  the 
notion  of  the  sum  of  the  angles  of  a  triangle,  and 
like  W'allis  he  relies  upon  the  idea  of  similitude, 
saying  that  "the  length  of  the  units  of  measurement 
is  indifferent  for  proving  the  theorems  in  ques- 
tion."^^ 

GAUSS. 

A  new  epoch  begins  with  Gauss,  or  rather  with 
his  ingenious  disciple  Riemann.  While  Gauss  was 
rather  timid  about  speaking  openly  on  the  subject, 
he  did  not  wish  his  ideas  to  be  lost  to  posterity.  In 
a  letter  to  Schumacher  dated  May  17,  1831,  he 
said: 

"I  have  bej^^un  to  jot  down  somethincT  of  my  own  medi- 
tations, which  are  partly  older  than  forty  years,  but  wliich 
I  have  never  written  out.  beings  obhp^ed  therefore  to  excogi- 
tate many  things  three  or  ff)ur  times  over.  I  do  not  wish 
them  to  pass  away  with  me." 

The  notes  to  which  Gauss  here  refers  have  not 
been  found  among  his  postluunous  papers,  and  il 

^''  Mnnnirrs  dc  l' Academic  dcs  Sciences  de  I'liistilut  de  !■  ranee. 
Vol.  XII,  1833. 


12  THE   FOUNDATIONS   OF   MATHEMATICS. 

therefore  seems  probable  that  they  are  lost,  and  our 
knowledge  of  his  thoughts  remains  limited  to  the 
comments  that  are  scattered  through  his  corres- 
pondence with  mathematical  friends. 

Gauss  wrote  to  Bessel  (1784- 1846)  January  27, 
1829: 

"T  have  also  in  my  leisure  hours  frequently  reflected  upon 
another  problem,  now  of  nearly  forty  years'  standing.  I 
refer  to  the  foundations  of  geometry.  I  do  not  know 
whether  I  have  ever  mentioned  to  you  my  views  on  this 
matter.  Mv  meditations  here  also  have  taken  more  definite 
shape,  and  my  conviction  that  we  cannot  thoroughly  demon- 
strate geometry  a  priori  is,  if  possible,  more  strongly  con- 
firmed than  ever.  But  it  will  take  a  long  time  for  me  to 
bring  myself  to  the  point  of  working  out  and  making  public 
my  very  extensizr  investigations  on  this  subject,  and  pos- 
sibly this  will  not  be  done  during  my  life,  inasmuch  as  I 
stand  in  dread  of  the  clamors  of  the  Boeotians,  which  would 
be  certain  to  arise,  if  I  should  ever  give  full  expression  to 
my  views.  It  is  curious  that  in  addition  to  the  celebrated 
flaw  in  Euclid's  Geometry,  which  mathematicians  have  hith- 
erto endeavored  in  vain  to  patch  and  never  will  succeed, 
there  is  still  another  blotch  in  its  fabric  to  which,  so  far  as 
I  know,  attention  has  never  yet  been  called  and  which  it  will 
by  no  means  be  easy,  if  at  all  possible,  to  remove.  This  is 
the  definition  of  a  plane  as  a  surface  in  which  a  straight 
line  joining  any  tzvo  points  lies  wholly  in  that  plane.  This 
definition  contains  more  than  is  requisite  to  the  determina- 
tion of  a  surface,  and  tacitly  involves  a  theorem  which  is  in 
need  of  prior  proof." 

Bessel  in  his  answer  to  Gauss  makes  a  distinc- 
tion between  Euclidean  geometry  as  practical  and 
metageometry  (the  one  that  does  not  depend  upon 


HISTORICAL  SKETCH.  '  I3 

the  theorem  of  parallel  lines)    as  true  geometry. 
He  writes  under  the  date  of  February  lo,  1829: 

"I  should  regard  it  as  a  great  misfortune  if  you  were  to 
allow  yourself  to  be  deterred  by  the  'clamors  of  the  Boeo- 
tians' from  explaining  your  views  of  geometry.  From  what 
Lambert  has  said  and  Schweikart  orally  communicated,  it 
has  become  clear  to  me  that  our  geometry  is  incomplete  and 
stands  in  need  of  a  correction  which  is  hypothetical  and 
which  vanishes  when  the  sum  of  the  angles  of  a  plane  tri- 
angle is  equal  to  180°.  This  would  be  the  true  geometry 
and  the  Euclidean  the  practical,  at  least  for  figures  on  the 
earth." 

In  another  letter  to  Bessel,  April  9,  1830,  Gauss 
sums  up  his  views  as  follows : 

"The  ease  with  which  you  have  assimilated  my  notions  of 
geometry  has  been  a  source  of  genuine  delight  to  me,  espe- 
cially as  so  few  possess  a  natural  bent  for  them.  I  am  pro- 
foundly convinced  that  the  theory  of  space  occupies  an  en- 
tirely different  position  with  regard  to  our  knowledge  a 
priori  from  that  ot  the  theory  of  numbers  (Grdssciilehre)  ; 
that  perfect  conviction  of  the  necessity  and  therefore  the 
absolute  truth  which  is  characteristic  of  the  latter  is  totally 
wanting  to  our  knowledge  of  the  former.  We  must  confess 
in  all  humility  that  a  number  is  solely  a  product  of  our  mind. 
Space,  on  the  other  hand,  possesses  also  a  reality  outside  of 
our  mind,  the  laws  of  which  we  cannot  fully  prescribe  a 
priori." 

Another  letter  of  Gauss  may  be  quoted  here  in 
full.  Tt  is  a  re])ly  to  Taurinus  and  contains  an  ap- 
preciation of  his  essay  on  the  Parallel  Lines.  Gauss 
writes  from  Gottingcn,  Nov.  (S,  i(S24: 

"Your  esteemed  communication  of  October  30th,  with 


14  THE   FOUNDATIONS   OF   MATHEMATICS. 

the  accompanying  little  essay,  I  have  read  with  considerable 
pleasure,  the  more  so  as  I  usually  find  no  trace  whatever  of 
real  geometrical  talent  in  the  majority  of  the  people  who 
ofTer  new  contributions  to  the  so-called  theory  of  parallel 
lines. 

"With  regard  to  your  effort,  I  have  nothing  (or  not 
much)  more  to  say,  except  that  it  is  incomplete.  Your  pres- 
entation of  the  demonstration  that  the  sum  of  the  three  an- 
gles of  a  plane  triangle  cannot  be  greater  than  i8o°,  does 
indeed  leave  something  to  be  desired  in  point  of  geometrical 
precision.  But  this  could  be  supplied,  and  there  is  no  doubt 
that  the  impossibility  in  question  admits  of  the  most  rigorous 
demonstration.  But  the  case  is  quite  different  with  the 
second  part,  viz.,  that  the  sum  of  the  angles  cannot  be 
smaller  than  i8o° ;  this  is  the  real  difficulty,  the  rock  on 
which  all  endeavors  are  wrecked.  I  surmise  that  you  have 
not  employed  yourself  long  with  this  subject.  I  have  pon- 
dered it  for  more  than  thirty  years,  and  I  do  not  believe 
that  any  one  could  have  concerned  himself  more  exhaus- 
tively with  this  second  part  than  I,  although  I  have  not 
published  anything  on  this  subject.  The  assumption  that 
the  sum  of  the  three  angles  is  smaller  than  i8o°  leads  to  a 
new  geometry  entirely  different  from  our  Euclidean, — a 
geometry  which  is  throughout  consistent  with  itself,  and 
which  I  have  elaborated  in  a  manner  entirely  satisfactory 
to  myself,  so  that  I  can  solve  every  problem  in  it  with  the 
exception  of  the  determining  of  a  constant,  which  is  not 
a  priori  obtainable.  The  larger  this  constant  is  taken,  the 
nearer  we  approach  tlie  Euclidean  geometry,  and  an  infin- 
itely large  value  will  make  the  two  coincident.  The  propo- 
sitions of  this  geometry  appear  partly  paradoxical  and  ab- 
surd to  the  uninitiated,  but  on  closer  and  calmer  considera- 
tion it  will  be  found  that  they  contain  in  them  absolutely 
nothing  that  is  impossible.  Thus,  the  three  angles  of  a 
triangle,  for  example,  can  be  made  as  small  as  we  will, 
provided  the  sides  can  be  taken  large  enough ;  whilst  the 


HISTORICAL  SKETCH.  15 

area  of  a  triangle,  however  great  the  sides  may  be  taken, 
can  never  exceed  a  definite  limit,  nay,  can  never  once  reach 
it.  All  my  endeavors  to  discover  contradictions  or  incon- 
sistencies in  this  non-Euclidean  geometry  have  been  in  vain, 
and  the  only  thing  in  it  that  conflicts  with  our  reason  is  the 
fact  that  if  it  were  true  there  would  necessarily  exist  in  space 
a  linear  magnitude  quite  determinate  in  itself,  yet  unknown 
to  us.  But  I  opine  that,  despite  the  empty  word-wisdom  of 
the  metaphysicians,  in  reality  we  know  little  or  nothing  of 
the  true  nature  of  space,  so  much  so  that  w^e  are  not  at  liberty 
to  characterize  as  absolutely  impossible  things  that  strike  us 
as  unnatural.  If  the  non-Euclidean  geometry  were  the  true 
geometry,  and  the  constant  in  a  certain  ratio  to  such  mag- 
nitudes as  lie  within  the  reach  of  our  measurements  on  the 
earth  and  in  the  heavens,  it  could  be  determined  a  posteriori. 
I  have,  therefore,  in  jest  frequently  expressed  the  desire  that 
the  Euclidean  geometry  should  not  be  the  true  geometry, 
because  in  that  event  we  should  have  an  absolute  measure 
a  priori." 

Schweikart,  a  contemporary  of  Gauss,  may  in- 
cidentally be  mentioned  as  having-  worked  out  a 
geometry  that  would  l)c  independent  of  the  Euclid- 
ean axiom.    He  called  il  astral  geometry.^^ 

RIKMANN. 

Gauss's  ideas  fell  upon  good  soil  in  his  disciple 
Riemann  (1826-1866)  whose  liahilitati/m  Lecture 
on  "The  li\j)otheses  which  Constilule  the  leases  of 
Ger)metry"  inaugurates  a  new  epoch  in  the  history 
of  the  philosophy  of  mathematics. 

Riemann  states  the  situation  as  follows.    T  c|uote 

Die  Thcorie  der  Parallclliuicn,  tichsi  dcm  I'orschlag  Hirer  I'er- 
bannung  aus  der  Geometric.     Lcipsic  and  Jena,  1807. 


l6  THE  FOUNDATIONS  OF   MATHEMATICS. 

from  Clifford's  almost  too  literal  translation  (first 
published  in  Nature,  1873) : 

"It  is  known  that  geometry  assumes,  as  things  given, 
both  the  notion  of  space  and  the  first  principles  of  construc- 
tions in  space.  She  gives  definitions  of  them  which  are 
merely  nominal,  while  the  true  determinations  appear  in  the 
form  of  axioms.  The  relation  of  these  assumptions  remains 
consequently  in  darkness  ;  we  neither  perceive  whether  and 
how  far  their  connection  is  necessary,  nor,  a  priori,  whether 
it  is  possible. 

"From  Euclid  to  Legendre  (to  name  the  most  famous  of 
modern  reforming  geometers)  this  darkness  was  cleared  up 
neither  by  mathematicians  nor  by  such  philosophers  as  con- 
cerned themselves  with  it." 

Riemann  arrives  at  a  conclusion  which  is  nega- 
tive.    He  says : 

"The  propositions  of  geometry  cannot  be  derived  from 
general  notions  of  magnitude,  but  the  properties  which  dis- 
tinguish space  from  other  conceivable  triply  extended  mag- 
nitudes are  only  to  be  deduced  from  experience." 

In  the  attempt  at  discovering  the  simplest  mat- 
ters of  fact  from  which  the  measure-relations  of 
space  may  be  determined,  Riemann  declares  that — 

"Like  all  matters  of  fact,  they  are  not  necessary,  but 
only  of  empirical  certainty  ;  they  are  hypotheses." 

Being  a  mathematician,  Riemann  is  naturally 
bent  on  deductive  reasoning,  and  in  trying  to  find 
a  foothold  in  the  emptiness  of  pure  abstraction  he 
starts  with  general  notions.  He  argues  that  posi- 
tion must  be  determined  by  measuring  quantities, 
and  this  necessitates  the  assumption  that  length  of 


HISTORICAL  SKETCH.  I7 

lines  is  independent  of  position.  Then  he  starts  with 
the  notion  of  manifoldness,  which  he  undertakes  to 
specialize.  This  speciahzation,  however,  may  be 
done  in  various  ways.  It  may  be  continuous,  as  is 
geometrical  space,  or  consist  of  discrete  units,  as 
do  arithmetical  numbers.  We  may  construct  mani- 
foldnesses  of  one,  two,  three,  or  h  dimensions,  and 
the  elements  of  which  a  system  is  constructed  may 
be  functions  which  undergo  an  infinitesimal  dis- 
placement expressible  by  dx.  Thus  spaces  become 
possible  in  which  the  directest  linear  functions  (an- 
alogous to  the  straight  lines  of  Euclid )  cease  to  be 
straight  and  sufifer  a  continuous  deflection  which 
may  be  positive  or  negative,  increasing  or  decreas- 
ing. 

Riemann  argues  that  the  simplest  case  will  be, 
if  the  differential  line-element  ds  is  the  square  root 
of  an  always  ])ositive  integral  homogeneous  function 
of  the  second  order  of  the  f|uantities  dx  in  which 
the  coefficients  are  continuous  functions  of  the  quan- 
tities X,  viz.,  ds  =\/tdx-,  but  it  is  one  instance  only 
of  a  whole  class  of  possibilities.     lie  savs: 

"Manifoldnesses  in  which,  as  in  tin-  plane  and  in  space, 
the  line-element  may  be  reduced  to  the  form  y^^dx^,  are 
therefore  only  a  particular  case  of  the  manifoldnesses  to  be 
here  investigated  :  they  require  a  special  name,  and  therefore 
these  manifoldnesses  in  which  the  square  of  the  line-element 
may  be  expressed  as  the  sum  of  the  squares  of  complete  dif- 
ferentials I  will  call  flat." 

The  Euclidean  plane  is  the  best-known  instance 
of  flat  space  being  a  manifold  f)f  ,-i  zero  curvatiu-c. 


l8  THE   FOUNDATIONS   OF   MATHEMATICS. 

Flat  or  even  space  has  also  been  called  by  the 
new-fangled  word  Jiomaloidal,^^  which  recommends 
itself  as  a  technical  term  in  distinction  from  the 
popular  meaning  of  even  and  flat. 

In  applying  his  determination  of  the  general 
notion  of  a  manifold  to  actual  space,  Riemann  ex- 
presses its  properties  thus : 

"In  the  extension  of  space-construction  to  the  infinitely 
great,  we  must  distinguish  between  nnbonndedness  and  in- 
finite extent ;  the  former  belongs  to  the  extent-relations,  the 
latter  to  the  measure  relations.  That  space  is  an  unbounded 
threefold  manifoldness,  is  an  assumption  which  is  developed 
by  every  conception  of  the  outer  world ;  according  to  which 
every  instant  the  region  of  real  perception  is  completed  and 
the  possible  positions  of  a  sought  object  are  constructed,  and 
which  by  these  applications  is  forever  confirming  itself.  The 
unboundedness  of  space  possesses  in  this  way  a  greater  em- 
pirical certainty  than  any  external  experience.  But  its  in- 
finite extent  by  no  means  follows  from  this ;  on  the  other 
hand,  if  we  assume  independence  of  bodies  from  position, 
and  therefore  ascribe  to  space  constant  curvature,  it  must 
necessarily  be  finite,  provided  this  curvature  has  ever  so 
small  a  positive  value.  If  we  prolong  all  the  geodetics 
starting  in  a  given  surface-element,  we  should  obtain  an 
unbounded  surface  of  constant  curvature,  i.  e.,  a  surface 
which  in  a  flat  manifoldness  of  three  dimensions  would  take 
the  form  of  a  sphere,  and  consequently  be  finite." 

It  is  obvious  from  these  quotations  that  Rie- 
mann is  a  disciple  of  Kant.  He  is  inspired  by  his 
teacher  Gauss  and  by  Herbart.  But  while  he  starts 
a  transcendentalist,  employing  mainly  the  method 
of  deductive  reasoning,  he  arrives  at  results  which 

"  From  the  Greek  6/uo\6s,  level. 


HISTORICAL  SKETCH.  IQ 

would  Stamp  him  an  empiricist  of  the  school  of  Mill. 
He  concludes  that  the  nature  of  real  space,  which  is 
only  one  instance  among  many  possibilities,  must 
be  determined  a  posteriori.  The  problem  of  tri- 
dimensionality  and  homaloidality  are  questions 
which  must  be  decided  by  experience,  and  while 
upon  the  whole  he  seems  inclined  to  grant  that 
Euclidean  geometry  is  the  most  practical  for  a  solu- 
tion of  the  coarsest  investigations,  he  is  inclined 
to  believe  that  real  space  is  non-Euclidean.  Though 
the  deviation  from  the  Euclidean  standard  can  only 
be  slight,  there  is  a  possibility  of  determining  it  by 
exact  measurement  and  observation. 

Riemann  has  succeeded  in  impressing  his  view 
upon  meta-geometricians  down  to  the  present  day. 
They  have  built  higher  and  introduced  new  ideas, 
yet  the  cornerstone  of  metageometry  remained  the 
same.  It  will  therefore  be  found  recommendable 
in  a  discussion  of  the  problem  to  begin  with  a  criti- 
cism of  his  Habilitation  Lecture. 

It  is  regrettable  that  Riemann  was  not  allowed  to 
work  out  his  philosophy  of  mathematics.  He  died 
at  the  premature  age  of  forty.  ])ut  the  work  which 
he  pursued  with  so  much  success  had  already  been 
taken  up  by  two  others,  Lobatchevsky  and  Bolvai, 
who,  each  in  his  own  way,  actually  contrived  a 
geometry  independent  of  the  theorem  of  ])arallels. 

It  is  perhaps  no  accident  that  the  two  independ- 
ent and  almost  simultaneous  inventors  of  a  non- 
Euclidean  geometry  are  original,  not  to  say  way- 
ward, characters  living  on  the  outskirts  of  Euro- 


20  THE  FOUNDATIONS  OF   MATHEMATICS. 

pean  civilization,  the  one  a  Russian,  the  other  a 
Magyar. 

LOBATCHEVSKY. 

Nicolai  Ivanovich  Lobatchevsky^''  was  born  Oc- 
tober 22  (Nov.  2  of  our  calendar),  1793,  in  the 
town  of  Makariev,  about  40  miles  above  Nijni  Nov- 
gorod On  the  Volga.  His  father  was  an  architect 
who  died  in  1797,  leaving  behind  a  widow  and  two 
small  sons  in  poverty.  At  the  gymnasium  Loba- 
tchevsky  was  noted  for  obstinacy  and  disobedience, 
and  he  escaped  expulsion  only  through  the  protec- 
tion of  his  mathematical  teacher,  Professor  Bartels, 
w^ho  even  then  recognized  the  extraordinary  talents 
of  the  boy.  Lobatchevsky  graduated  with  distinc- 
tion and  became  in  his  further  career  professor  of 
mathematics  and  in  1827  Rector  of  the  University 
of  Kasan.  Two  books  of  his  offered  for  official 
publication  were  rejected  by  the  paternal  govern- 
ment of  Russia,  and  the  manuscripts  may  be  con- 
sidered as  lost  for  good.  Of  his  several  essays  on 
the  theories  of  parallel  lines  we  mention  only  the 
one  which  made  him  famous  throughout  the  whole 
mathematical  world,  Geometrical  Researches  on  the 
Theory  of  Parallels,  published  by  the  University  of 
Kasan  in  1835.^^ 

Lobatchevsky  divides  all  lines,  which  in  a  plane 
go  out  from  a  point  A  with  reference  to  a  given 

"  The  name  is  spelled  differently  according  to  the  different 
methods  of  transcribing  Russian  characters. 

"  For  further  details  see  Prof.  G.  B.  Halsted's  article  "Loba- 
chevski"  in  The  Open  Court,  1898,  pp.  411  ff. 


HISTORICAL  SKETCH. 


21 


Straight  line  BC  in  the  same  plane,  into  two  classes 
— cutting  and  not  cutting.  In  progressing  from  the 
not-cutting  lines,  such  as  EA  and  GA,  to  the  cutting 
lines,  such  as  FA,  we  must  come  upon  one  HA 
that  is  the  boundary  betw^een  the  two  classes ;  and  it 
is  this  which  he  calls  the  parallel  line.  He  desig- 
nates the  parallel  angle  on  the  perpendicular  {p^ 
AD,  dropped  from  A  upon  BC)  by  n.  If  n(/))  <i/<ir 
(viz.,  90  degrees)  we  shall  have  on  the  other  side 
of  p  another  angle  DAK=n(/))  parallel  to  DB, 
so  that  on  this  assumption  we 
must  make  a  distinction  of 
sides  in  parallelism,  and  we 
must  allow  two  parallels,  one 
on  the  one  and  one  on  the 
other  side.  If  n(/))  =  i/47r 
we  have  only  intersecting 
lines  and  one  parallel;  but  if 
U(p)<y2ir  we  have  two  par- 
allel lines  as  boundaries  be- 
tween the  intersecting  and 
non-intersecting  lines. 

W'c  need  not  further  develop  Lobatchevsky's 
idea.  Among  other  things,  he  proves  that  "if  in 
any  rectilinear  triangle  llie  sum  of  tlie  three  angles 
is  ecjual  to  two  right  angles,  so  is  this  also  the  case 
for  every  other  triangle,"  tliat  is  to  say.  each  in- 
stance is  a  sample  of  the  whok-.  and  if  one  case  is 
establislu'd,  the  nature  of  ihc  \\liole  system  to  which 
it  l)elongs  is  determined. 

The    importance    ot     Lobatchex'skv's    discoverv 


22  THE   FOUNDATIONS   OF   MATHEMATICS. 

consists  in  the  fact  that  the  assumption  of  a  geom- 
etry from  which  the  parallel  axiom  is  rejected,  does 
not  lead  to  self-contradictions  but  to  the  conception 
of  a  general  geometry  of  which  the  Euclidean  is  one 
possibility.  This  general  geometry  was  later  on 
most  appropriately  called  by  Lobatchevsky  "Pan- 
geometry." 

BOLYAI. 

John  (or,  as  the  Hungarians  say,  Janos)  Bolyai 
imbibed  the  love  of  mathematics  in  his  father's 
house.  He  was  the  son  of  Wolfgang  (or  Farkas) 
Bolyai,  a  fellow  student  of  Gauss  at  Gottingen  when 
the  latter  was  nineteen  years  old.  Farkas  was  pro- 
fessor of  mathematics  at  Maros  Vasarhely  and 
wrote  a  two-volume  book  on  the  elements  of  mathe- 
matics^'^ and  in  it  he  incidentally  mentions  his  vain 
attempts  at  proving  the  axiom  of  parallels.  His 
book  was  only  partly  completed  when  his  son  Janos 
wrote  him  of  his  discovery  of  a  mathematics  of  pure 
space.    He  said: 

"As  soon  as  I  have  put  it  into  order,  I  intend  to  write 
and  if  possible  to  publish  a  work  on  parallels.  At  this 
moment,  it  is  not  yet  finished,  but  the  way  which  I  have  fol- 
lowed promises  me  with  certainty  the  attainment  of  my  aim, 
if  it  is  at  all  attainable.  It  is  not  yet  attained,  but  I  have 
discovered  such  magnificent  things  that  I  am  myself  aston- 
ished at  the  result.  It  would  forever  be  a  pity,  if  they  were 
lost.    When  you  see  them,  my  father,  you  yourself  will  con- 

"  Tentamcn  juventutem  studiosam  in  elementa  matheos  etc.  iii- 
troducendi, printed  in  Maros  Vasarhely.  By  Farkas  Bolyai.  Part  I. 
Maros  Vasarhely,  1832.  It  contains  the  essay  by  Janos  Bolyai  as  an 
Appendix. 


HISTORICAL  SKETCH.  2^ 

cede  it.  Xow  I  cannot  say  more,  only  so  much  that  from 
nothing  I  haz'e  created  another  wholly  new  world.  All  that 
I  have  hitherto  sent  you  compares  to  it  as  a  house  of  cards 
to  a  castle."^® 

Janos  being  convinced  of  the  futility  of  proving 
Euclid's  axiom,  constructed  a  geometry  of  absolute 
space  which  would  be  independent  of  the  axiom  of 
parallels.  And  he  succeeded.  He  called  it  the  Sci- 
ence Absolute  of  Space}''  an  essay  of  twenty-four 
pages  which  Bolyai's  father  incorporated  in  the 
first  volume  of  his  Tentamen  as  an  appendix. 

Bolyai  was  a  thorough  Magyar.  He  was  wont 
to  dress  in  high  boots,  short  wide  Hungarian  trou- 
sers, and  a  white  jacket.  He  loved  the  violin  and 
was  a  good  shot.  \Miile  serving  as  an  of^cer  in 
the  Austrian  army,  Janos  was  known  for  his  hot 
temper,  which  finally  forced  him  to  resign  his  com- 
mission as  a  captain,  and  we  learn  from  Professor 
Halsted  that  for  some  provocation  he  was  chal- 
lenged by  thirteen  cavalry  officers  at  once.  Janos 
calmly  accepted  and  proposed  to  fight  them  all,  one 
after  the  other,  on  condition  that  he  be  permitted 
after  each  duel  to  play  a  piece  on  his  violin.  We 
know  not  the  nature  of  these  duels  nor  the  construc- 
tion of  the  pistols,  but  the  fact  remains  assured  that 
lie  came  out  unhurt.  As  for  the  rest  of  the  report 
that  "he  came  out  victor  from  the  thirteen  duels, 

"  Sec  Halsted's  introduction  to  the  English  translation  of  Bo- 
lyai's  Science  Absolute  of  Space,  p.  xxvii. 

^''Appendix  scicniiam  spalii  absolute  veram  cxhihcns:  a  vcrilate 
aut  falsitafe  axininatis  XI.  liuclidci  (a  priori  liaud  unquam  dcci- 
druda)  indcpcudcnicm ;  Adjccta  ad  casum  falsilatis  quadiatura  cir- 
culi  gcomelrica. 


24  THE   FOUNDATIONS   OF   MATHEMATICS. 

leaving  his  thirteen  adversaries  on  the  square," 
we  may  he  permitted  to  express  a  mild  but  deep- 
seated  doubt. 

janos  Bolyai  starts  with  straight  lines  in  the 
same  plane,  which  may  or  may  not  cut  each  other. 
Now  there  are  two  possibilities:  there  may  be  a 
system  in  which  straight  lines  can  be  drawn  which 
do  not  cut  one  another,  and  another  in  which  they  all 
cut  one  another.  The  former,  the  Euclidean  he 
calls  t,  the  latter  S.  "All  theorems,"  he  says, 
"which  are  not  expressly  asserted  for  S  or  for  S 
are  enunciated  absolutely,  that  is,  they  are  true 
whether  X  or  S  is  reality."^^  The  system  S  can 
he  established  without  axioms  and  is  actualized  in 
spherical  trigonometry,  (ibid.  p.  21).  Now  S  can 
be  changed  to  ,  viz.,  plane  geometry,  by  reducing 
the  constant  i  to  its  limit  (where  the  sect  y  =  o) 
which  is  practically  the  same  as  the  construction 
of  a  circle  with  r  ^  <^ ,  thus  changing  its  periphery 
into  a  straight  line. 

LATER    GEOMETRICIANS. 

The  labors  of  Lobatchevsky  and  Bolyai  are  sig- 
nificant in  so  far  as  they  prove  beyond  the  shadow 
of  a  doubt  that  a  construction  of  geometries  other 
than  Euclidean  is  possible  and  that  it  involves  us 
in  no  absurdities  or  contradictions.  This  upset  the 
traditional  trust  in  Euclidean  geometry  as  absolute 
truth,  and  it  opened  at  the  same  time  a  vista  of  new 

"  See  Halsted's  translation,  p.  14. 


HISTORICAL  SKETCH.  25 

problems,  foremost  among  which  was  the  (luestion 
as  to  the  mutual  relation  of  these  three  different 
geometries. 

It  was  Cayley  who  projiosed  an  answ-er  which 
was  further  elaborated  by  Felix  Klein.  These  two 
ingenious  mathematicians  succeeded  in  deriving  by 
projection  all  three  systems  from  one  common  ab- 
original form  called  by  Klein  Grundgebild  or  the 
Absolute.  In  addition  to  the  three  geometries  hith- 
erto known  to  mathematicians,  Klein  added  a  fourth 
one  which  he  calls  elliptic}^ 

Thus  w^e  may  now  regard  all  the  different  ge- 
ometries as  three  species  of  one  and  the  same  genus 
and  we  have  at  least  the  satisfaction  of  knowing  that 
there  is  terra  Urina  at  the  bottom  of  our  mathemat- 
ics, though  it  lies  deeper  than  was  formerly  sup- 
posed. 

Prof.  Simon  Newcomb  of  Johns  Hopkins  Uni- 
versity, although  not  familiar  with  Klein's  essays, 
worked  along  the  same  line  and  arrived  at  similar 
results  in  his  article  on  "Elementary  Theorems  Re- 
lating to  the  Geometry  of  a  Space  of  Three  Dimen- 
sions and  of  Uniform  Positive  Curvature  in  the 
Fourth  Dimension.^" 

In  the  meantime  the  problem  of  geometry  be- 
came interesting  to  outsiders  also,  for  the  theorem 
of  parallel   lines  is  a  problem  of  space.     A  most 

"  "Ucbcr  flic  sogenanntc  niclit-cuklidisclie  Geometric"  in  Math. 
Annalcn,  4,  6  (1X71-1872).  Vorlcsungcn  iibcr  nicht-euklidische  Geo- 
metric, Gottingen,  1893. 

"Crelle's  Journal  fur  die   reinc   und  duf^ezcaiidte   Matheinatik. 


26  THE  FOUNDATIONS  OF   MATHEMATICS. 

excellent  treatment  of  the  subject  came  from  the 
pen  of  the  great  naturalist  Helmholtz  who  wrote 
two  essays  that  are  interesting-  even  to  outsiders 
because  written  in  a  most  popular  style. ^^ 

A  collection  of  all  the  materials  from  Euclid  to 
Gauss,  compiled  by  Paul  Stackel  and  Friedrich 
Engel  under  the  title  Die  Thcorie  der  ParallelUnien 
von  Euklid  bis  auf  Gauss,  eine  Urkundensammlung 
^ur  Vorgeschichte  der  nielif-euklidischen  Geometrie, 
is  perhaps  the  most  useful  and  important  publica- 
tion in  this  line  of  thought,  a  book  which  has  become 
indispensable  to  the  student  of  metageometry  and 
its  history. 

A  store  of  information  may  be  derived  from 
Bertrand  A.  W.  Russell's  essay  on  the  Foundations 
of  Geometry.  He  divides  the  history  of  metageom- 
etry into  three  periods:  The  synthetic,  consisting 
of  suggestions  made  by  Legendre  and  Gauss;  the 
metrical,  inaugurated  by  Riemann  and  character- 
ized by  Lobatchevsky  and  Bolyai;  and  the  projec- 
tive, represented  by  Cayley  and  Klein,  who  reduce 
metrical  properties  to  projection  and  thus  show  that 
Euclidean  and  non-Euclidean  systems  may  result 
from  "the  absolute." 

Among  American  writers  no  one  has  contrib- 
uted more  to  the  interests  of  metageometry  than  the 
indefatigable  Dr.  George  Bruce  Halsted.^^    He  has 

"  "Ueber  die  thatsachlichen  Grundlagen  der  Geometrie,"  in  Wis- 
scnschaftl.  Ahh..,  1866,  Vol.  11.,  p.  610  ff..  and  "Ueber  die  Thatsachen, 
die  der  Geometrie  zu  Grunde  liegen,"  ibid.,  1868,  p.  618  ff. 

^  From  among  his  various  publications  we  mention  only  his 
translations :  Geometrical  Researches  on  the  Theory  of  Parallels  by 
Nicholaus  Lobatcheivsky.     Translated  from  the  Original.     And  The 


HISTORICAL  SKETCH.  27 

not  only  translated  Bolyai  and  Lobatchevsky,  but 
in  numerous  articles  and  lectures  advanced  his  own 
theories  toward  the  solution  of  the  problem. 

Prof.  B.  J.  Delboeuf  and  Prof.  H.  Poincare  have 
expressed  their  conceptions  as  to  the  nature  of  the 
bases  of  mathematics,  in  articles  contributed  to  The 
MonistP  The  latter  treats  the  subject  from  a  purely 
mathematical  standpoint,  while  Dr.  Ernst  Mach  in 
his  little  book  Space  and  Geometry, ^^  in  the  chapter 
'.'On  Physiological,  as  Distinguished  from  Geomet- 
rical, Space,"  attacks  the  problem  in  a  very  original 
manner  and  takes  into  consideration  mainly  the  nat- 
ural growth  of  space  conception.  His  exposition 
might  be  called  "the  physics  of  geometry." 

GRASSMANN. 

T  cannot  conclude  this  short  sketch  of  the  history 
of  metageometry  without  paying  a  tribute  to  the 
memory  of  Hermann  Grassmann  of  Stettin,  a  math- 
ematician of  first  degree  whose  highly  important 
results  in  this  line  of  work  have  only  of  late  found 

Science  Absolute  of  Space,  Independent  of  the  Truth  or  Falsity  of 
Euclid's  Axiom  XI .  (n-hich  can  never  be  decided  a  priori).  By  John 
Bolyai.  Translated  from  the  Latin,  both  publislied  in  Austin,  Texas, 
the  translator's  former  place  of  residence.  Further,  we  refer  the 
reader  to  Halsted's  bibliography  of  the  literature  on  hyperspace  and 
non-Ruclidean  geometry  in  the  American  Journal  of  Mathematics, 
Vol.  L,  pp.  261-276,  384,  385,  and  Vol.  IL,  pp.  65-70. 

"They  are  as  follows:  "Are  the  Dimensions  of  the  Physical 
World  Absolute?"  by  Prof.  B.  J.  Delbcruf,  The  Monist,  January, 
1894;  "On  the  Foundations  of  Geometry,"  by  Prof.  H.  Poincare, 
The  Monist,  Octolier,  1898;  also  "Relations  Between  Experimental 
and  Mathematical  Physics,"  The  Monist,  July,  1902. 

''Chicago:  Open  Court  Pub.  Co.,  1906.  This  chapter  also  ap- 
peared in  7  he  Mottist  for  Ai)ril,  1901. 


28  THE  FOUNDATIONS  OF  MATHEMATICS. 

the  recognition  which  they  so  fully  deserve.  I  do 
not  hesitate  to  say  that  Hermann  Grassmann's  Li- 
near e  Ausdehnungslehre  is  the  best  work  on  the 
philosophical  foundation  of  mathematics  from  the 
standpoint  of  a  mathematician. 

Grassmann  establishes  first  the  idea  of  mathe- 
matics as  the  science  of  pure  form.  He  shows  that 
the  mathematician  starts  from  definitions  and  then 
proceeds  to  show  how  the  product  of  thought  may 
originate  either  by  the  single  act  of  creation,  or  by 
the  double  act  of  positing  and  combining.  The 
former  is  the  continuous  form,  or  magnitude,  in  the 
narrower  sense  of  the  term,  the  latter  the  discrete 
form  or  the  method  of  combination.  He  distin- 
guishes between  intensive  and  extensive  magnitude 
and  chooses  as  the  best  example  of  the  latter  the 
sect^^  or  limited  straight  line  laid  down  in  some 
definite  direction.  Hence  the  name  of  the  new  sci- 
ence, "theory  of  linear  extension." 

Grassmann  constructs  linear  formations  of 
which  systems  of  one,  two,  three,  and  n  degrees 
are  possible.  The  Euclidean  plane  is  a  system  of 
second  degree,  and  space  a  system  of  third  degree. 
He  thus  generalizes  the  idea  of  mathematics,  and 
having  created  a  science  of  pure  form,  points  out 
that  geometry  is  one  of  its  applications  which  origi- 
nates under  definite  conditions. 

Grassmann  made  the  straight  line  the  basis  of 

^  Grassmann's  term  is  Strecke,  a  word  connected  with  the  Anglo- 
Saxon  "Stretch,"  being  that  portion  of  a  hne  that  stretches  between 
two  points.  The  translation  "sect,"  was  suggested  by  Prof.  G.  B. 
Halsted. 


HISTORICAL  SKETCH.  29 

his  geometrical  definitions.  He  defines  the  plane 
as  the  totality  of  parallels  which  cut  a  straight  line 
and  space  as  the  totality  of  parallels  which  cut  the 
plane.  Here  is  the  limit  to  geometrical  construction, 
hut  abstract  thought  knows  of  no  bounds.  Having 
generalized  our  mathematical  notions  as  systems 
of  first,  second,  and  third  degree,  we  can  continue 
in  the  numeral  series  and  construct  systems  of  four, 
five,  and  still  higher  degrees.  Further,  we  can  de- 
termine any  plane  by  any  three  points,  given  in  the 
figures  Xi,  X2,  X:i,  not  lying  in  a  straight  line.  If  the 
equation  between  these  three  figures  be  homogene- 
ous, the  totality  of  all  points  that  correspond  to  it  will 
be  a  system  of  second  degree.  If  this  homogeneous 
equation  is  of  the  first  grade,  this  system  of  second 
degree  will  be  simple,  viz.,  of  a  straight  line;  but 
if  the  equation  be  of  a  higher  grade,  we  shall  have 
curves  for  which  not  all  the  laws  of  plane  geometry 
hold  good.  The  same  considerations  lead  to  a  dis- 
tinction between  homaloidal  space  and  non-Euclid- 
ean systems.^" 

Being  professor  at  a  German  gymnasium  and 
not  a  university,  Grassniann's  1)ook  remained  neg- 
Icclcd  and  the  newness  of  liis  methods  prevented 
superfici.'d  readers  from  ap])reci.'iling  the  sweeping 
significance  of  his  ])ropositions.  Since  there  was 
no  call  whatever  fnr  ilie  book,  the  publishers  re- 
turned the  whole  edition  to  the  i)aper  mill,  and  the 
complimentary  copies   whieli    the   autlior   bad   sent 

**  Sec  Gras<;mann's  Ausdclinuiiiislclirr.   1844,  AiiliaiiK   i.  pI'-  -'7.V 
274. 


30  THE   FOUNDATIONS  OF   MATHEMATICS. 

out  to  his  friends  are  perhaps  the  sole  portion  that 
was  saved  from  the  general  doom. 

Grassmann,  disappointed  in  his  mathematical 
labors,  had  in  the  meantime  turned  to  other  studies 
and  gained  the  honorary  doctorate  of  the  Univer- 
sity of  Tubingen  in  recognition  of  his  meritorious 
work  on  the  St.  Petersburg  Sankrit  Dictionary, 
when  Victor  Schlegel  called  attention  to  the  simi- 
larity of  Hamilton's  theory  of  vectors  to  Grass- 
mann's  concept  oiStrccke,  both  being  limited  straight 
lines  of  definite  direction.  Suddenly  a  demand  for 
Grassmann's  book  was  created  in  the  market;  but 
alas!  no  copy  could  be  had,  and  the  publishers 
deemed  it  advisable  to  reprint  the  destroyed  edition 
of  1844.  The  satisfaction  of  this  late  recognition 
was  the  last  joy  that  brightened  the  eve  of  Grass- 
mann's life.  He  wrote  the  introduction  and  an  ap- 
pendix to  the  second  edition  of  his  Lineare  Aus- 
dehmingslehre,  but  died  while  the  forms  of  his  book 
were  on  the  press. 

At  the  present  day  the  literature  on  metageo- 
metrical  subjects  has  grown  to  such  an  extent  that 
we  do  not  venture  to  enter  into  further  details.  We 
will  only  mention  the  appearance  of  Professor 
Schoute's  work  on  more-dimensional  geometry^^ 
which  promises  to  be  the  elaboration  of  the  pan- 
geometrical  ideal. 

"  Mehrdimensionale  Geometrie  von  Dr.  P.  H.  Schoute,  Professor 
der  Math,  an  d.  Reichs-Universitat  zu  Groningen,  Holland.  Leipsic. 
Goschen.  So  far  only  the  first  volume,  which  treats  of  linear  space, 
has  appeared. 


HISTORICAL  SKETCH.  3I 

EUCLID    STILL    L'^NIMPAIRED. 

Having  briefly  examined  the  chief  innovations 
of  modern  times  in  the  field  of  elementary  geometry, 
it  ought  to  be  pointed  out  that  in  spite  of  the  well- 
deserved  fame  of  the  metageometricians  from  Wal- 
lis  to  Halsted,  Euclid's  claim  to  classicism  remains 
unshaken.  The  metageometrical  movement  is  not 
a  revolution  against  Euclid's  authority  but  an  at- 
tempt at  widening  our  mathematical  horizon.  Let 
us  hear  what  Halsted,  one  of  the  boldest  and  most 
iconoclastic  among  the  chamj^ions  of  metageometry 
of  the  present  day,  has  to  say  of  Euclid.  Halsted 
begins  the  Introduction  to  his  English  translation 
of  Bolyai's  Science  Absolute  of  Space  with  a  terse 
description  of  the  history  of  Euclid's  great  book 
TJie  Elements  of  Geometry,  the  rediscovery  of  which 
is  not  the  least  factor  that  initiated  a  new  epoch  in 
the  development  of  Europe  which  may  be  called  the 
era  of  inventions,  of  discoveries,  and  of  the  appre- 
ciation as  well  as  growth  of  science.    Halsted  says: 

"The  immortal  Rlcmcnts  of  Euclid  was  already  in  dim 
antif|uity  a  classic.  re£:^ardcfl  as  absolutely  perfect,  valid 
without  restriction. 

"Elementary  geometry  was  for  two  thousand  years  as 
stationary,  as  fixed,  as  peculiarly  Greek  as  the  Parthenon. 
Dn  this  founflation  j)ure  science  rose  in  .Archimedes,  in 
Apollonius,  in  Pappus;  strugpfled  in  Thcon.  in  fTypatia; 
declined  in  Proclus  ;  fell  into  the  lon.e;-  decadence  of  (he  Dark 
Aji^cs. 

"The  l)of»k  that  monkish  luirope  could  no  longer  under- 


32  THE  FOUNDATIONS  OF  MATHEMATICS. 

Stand  was  then  taught  in  Arabic  by  Saracen  and  Moor  in 
the  Universities  of  Bagdad  and  Cordova. 

"To  bring  the  light,  after  weary,  stupid  centuries,  to 
Western  Christendom,  an  Englishman,  Adelhard  of  Bath, 
journeys,  to  learn  Arabic,  through  Asia  Minor,  through 
Egypt,  back  to  Spain,  Disguised  as  a  Mohammedan  stu- 
dent, he  got  into  Cordova  about  1120,  obtained  a  Moorish 
copy  of  Euclid's  Elements,  and  made  a  translation  from  the 
Arabic  into  Latin. 

"The  first  printed  edition  of  Euclid,  published  in  Venice 
in  1482,  was  a  Latin  version  from  the  Arabic.  The  trans- 
lation into  Latin  from  the  Greek,  made  by  Zamberti  from  a 
manuscript  of  Theon's  revision,  was  first  published  at  Ven- 
ice in  1505. 

"Twenty-eight  years  later  appeared  the  edifio  prince ps  in 
Greek,  published  at  Basle  in  1533  by  John  Hervagius, 
edited  by  Simon  Grynaeus.  This  was  for  a  century  and 
three-quarters  the  only  printed  Greek  text  of  all  the  books, 
and  from  it  the  first  English  translation  (1570)  was  made 
by  'Henricus  Billingsley,'  afterward  Sir  Henry  Billingsley, 
Lord  Mayor  of  London  in  1591. 

"And  even  to-day,  1895,  in  the  vast  system  of  examina- 
tions carried  out  by  the  British  Government,  by  Oxford, 
and  by  Cambridge,  no  proof  of  a  theorem  in  geometry  will 
be  accepted  which  infringes  Euclid's  sequence  of  propo- 
sitions. 

"Nor  is  the  work  unworthy  of  this  extraordinary  im- 
mortality. 

"Says  CliflPord:  'This  book  has  been  for  nearly  twenty- 
two  centuries  the  encouragement  and  guide  of  that  scientific 
thought  which  is  one  thing  with  the  progress  of  man  from 
a  worse  to  a  better  state. 

"  'The  encouragement :  for  it  contained  a  body  of  knowl- 
edge that  was  really  known  and  could  be  relied  on. 

"  'The  guide ;  for  the  aim  of  every  student  of  every  sub- 


HISTORICAL  SKETCH.  33 

ject  was  to  brin,^  his  knowledge  of  that  subject  into  a  form 
as  perfect  as  that  which  geometry  had  attained.'  " 

Euclid's  Elements  of  Geometry  is  not  counted 
among  the  books  of  divine  revelation,  but  truly  it 
deserves  to  be  held  in  religious  veneration.  There 
is  a  real  sanctity  in  mathematical  truth  which  is 
not  sufficiently  appreciated,  and  certainly  if  truth, 
helpfulness,  and  directness  and  simplicity  of  presen- 
tation, give  a  title  to  rank  as  divinely  inspired  litera- 
ture, Euclid's  great  work  should  be  counted  among 
the  canonical  books  of  mankind. 


Is  there  any  need  of  warning  our  readers  that 
the  foregoing  sketch  of  the  history  of  metageometry 
is  both  brief  and  popular?  We  have  purposely 
avoided  the  discussion  of  technical  details,  limiting 
our  exposition  to  the  most  essential  points  and  try- 
ing to  show  them  in  a  light  that  will  render  them 
interesting  even  to  the  non-mathematical  reader. 
It  is  meant  to  serve  as  an  introduction  to  the  real 
matter  in  hand,  viz.,  an  examination  of  the  founda- 
tions upon  which  geometrical  truth  is  to  be  ration- 
ally justified. 

The  author  has  purposely  introduced  wliat 
nu'ght  be  called  a  biographical  element  in  these  ex- 
positions of  a  subject  which  is  commonly  regarded 
as  dry  and  abstruse,  and  endeavored  to  give  some- 
thing of  the  lives  of  the  men  who  have  struggled 
and  labored  in  this  line  of  thought  and  have  sacri- 
ficed their  time  anfl  energy  on  the  altar  of  one  of  tlic 


34  THE   FOUNDATIONS   OF   MATHEMATICS. 

noblest  aspirations  of  man,  the  delineation  of  a 
philosophy  of  mathematics.  He  hopes  thereby  to 
relieve  the  dryness  of  the  subject  and  to  create  an 
interest  in  the  labor  of  these  pioneers  of  intellectual 
progress. 


THE  PHILOSOPHICAL  BASLS  OF  ^lATHE- 
MATICS. 

THE   PHILOSOPHICAL   PROBLEM. 

HAVING  thus  reviewed  the  history  of  non- 
EucHdean  geometry,  which,  rightly  consid- 
ered, is  but  a  search  for  the  philosophy  of  mathe- 
matics, I  now  turn  to  the  problem  itself  and,  in  the 
conviction  that  I  can  offer  some  hints  which  con- 
tain its  solution,  I  will  formulate  my  own  views  in 
as  popular  language  as  w^ould  seem  compatible  with 
exactness.  Not  being  a  mathematician  by  profes- 
sion I  have  only  one  excuse  to  offer,  which  is  this: 
that  T  have  more  and  more  come  to  the  conclusion 
that  the  problem  is  not  mathematical  but  philo- 
sophical ;  and  I  hope  that  those  who  are  competent 
to  judge  will  correct  me  where  I  am  mistaken. 

The  problem  of  the  philosophical  foundation  of 
mathematics  is  closely  connected  with  tlic  topics 
of  Kant's  Critique  of  Pure  Reason.  It  is  the  old 
quarrel  between  Empiricism  and  Transcendental- 
ism. Hence  our  method  of  dealing  with  it  will  nat- 
urally be  philosophical,  not  typically  mathematical. 

The  proper  solution  can  be  attained  only  by 
analysing  the  fundamental  concepts  of  mathematics 


36  THE  FOUNDATIONS  OF   MATHEMATICS. 

and  by  tracing  them  to  their  origin.  Thus  alone  can 
we  know  their  nature  as  well  as  the  field  of  their 
applicability. 

We  shall  see  that  the  data  of  mathematics  are 
not  without  their  premises;  they  are  not,  as  the 
Germans  say,  Z'oratissetsuugslos ;  and  though  math- 
ematics is  built  up  from  nothing,  the  mathematician 
does  not  start  with  nothing.  He  uses  mental  im- 
plements, and  it  is  they  that  give  character  to  his 
science. 

Obviously  the  theorem  of  parallel  lines  is  one 
instance  only  of  a  difficulty  that  betrays  itself  every- 
where in  various  forms;  it  is  not  the  disease  of 
geometry,  but  a  symptom  of  the  disease.  The  the- 
orem that  the  sum  of  the  angles  in  a  triangle  is 
equal  to  180  degrees;  the  ideas  of  the  evenness  or 
homaloidality  of  space,  of  the  rectangularity  of  the 
square,  and  more  remotely  even  the  irrationality  of 
TT  and  of  e,  are  all  interconnected.  It  is  not  the 
author's  intention  to  show  their  interconnection, 
nor  to  prove  their  interdependence.  That  task  is 
the  work  of  the  mathematician.  The  present  in- 
vestigation shall  be  limited  to  the  philosophical  side 
of  the  problem  for  the  sake  of  determining  the  na- 
ture of  our  notions  of  evenness,  which  determines 
both  parallelism  and  rectangularity. 

At  the  bottom  of  the  difficulty  there  lurks  the  old 
problem  of  apriority,  proposed  by  Kant  and  decided 
by  him  in  a  way  which  promised  to  give  to  mathe- 
matics a  solid  foundation  in  the  realm  of  transcen- 
dental thought.    And  yet  the  transcendental  method 


THE  PHILOSOPHICAL  BASIS.  37 

finally  sent  geometry  away  from  home  in  search  of 
a  new  domicile  in  the  wide  domain  of  empiricism. 

Riemann,  a  disciple  of  Kant,  is  a  transcendental- 
ist.  He  starts  with  general  notions  and  his  arguments 
are  deductive,  leading  him  from  the  abstract  down 
to  concrete  instances;  but  when  stepping  from  the 
ethereal  height  of  the  absolute  into  the  region  of 
definite  space-relations,  he  fails  to  find  the  necessary 
connection  that  characterizes  all  a  priori  reasoning; 
and  so  he  swerves  into  the  domain  of  the  a  posteriori 
and  declares  that  the  nature  of  the  specific  features 
of  space  must  be  determined  by  experience. 

The  very  idea  seems  strange  to  those  who  have 
been  reared  in  traditions  of  the  old  school.  An  un- 
sophisticated man,  when  he  speaks  of  a  straight 
line,  means  that  straightness  is  implied  thereby; 
and  if  he  is  told  that  space  may  be  such  as  to  render 
all  straightest  lines  crooked,  he  will  naturally  be 
ijewildered.  If  his  metageometrical  friend,  with 
much  learnedness  and  in  sober  earnest,  tells  him  that 
when  he  sends  out  a  ray  as  a  straight  line  in  a  for- 
ward direction  it  will  imperceptibly  deviate  and 
finally  turn  back  upon  his  occiput,  he  will  naturally 
become  suspicious  of  the  mental  soundness  of  his 
company.  \\'ould  not  many  of  us  dismiss  such  ideas 
wuth  a  shrug  if  there  were  not  geniuses  of  the  very 
first  rank  who  subscribe  to  the  same?  So  in  all 
modesty  we  have  to  defer  our  judgment  until  com- 
petent study  and  mature  reflection  have  enabled  us 
to  understand  the  difificulty  which  they  encounter 
and  then  judge  their  solution.     One  thing  is  sure, 


38  THE  FOUNDATIONS  OF   MATHEMATICS. 

however:  if  there  is  anything  wrong  with  meta- 
geometry,  the  fault  Hes  not  in  its  mathematical 
portion  but  must  be  sought  for  in  its  philosophical 
foundation,  and  it  is  this  problem  to  which  the 
present  treatise  is  devoted. 

While  we  propose  to  attack  the  problem  as  a 
philosophical  question,  we  hope  that  the  solution 
will  prove  acceptable  to  mathematicians. 

TRANSCENDENTALISM    AND   EMPIRICISM. 

In  philosophy  we  have  the  old  contrast  between 
the  empiricist  and  transcendentalist  school.  The 
former  derive  everything  from  experience,  the  latter 
insist  that  experience  depends  upon  notions  not  de- 
rived from  experience,  called  transcendental,  and 
these  notions  are  a  priori.  The  former  found  their 
representative  thinkers  in  Locke,  Hume,  and  John 
Stuart  Mill,  the  latter  was  perfected  by  Kant.  Kanf 
establishes  the  existence  of  notions  of  the  a  priori 
on  a  solid  basis  asserting  their  universality  and 
necessity,  but  he  no  longer  identified  the  a  priori 
with  innate  ideas.  He  granted  that  much  to  em- 
piricism, stating  that  all  knowledge  begins  with  ex- 
perience and  that  experience  rouses  in  our  mind 
the  a  priori  which  is  characteristic  of  mind.  Mill 
went  so  far  as  to  deny  altogether  necessity  and  uni- 
versality, claiming  that  on  some  other  planet  2X2 
might  be  5.  French  positivism,  represented  by 
Comte  and  IJttre,  follows  the  lead  of  Mill  and  thus 
they  end  in  agnosticism,  and  the  same  result  was 


THE  PHILOSOPHICAL  BASIS.  39 

reached  in  England  on  grounds  somewhat  different 
by  Herbert  Spencer. 

The  way  which  we  propose  to  take  may  be  char- 
acterized as  the  New  Positivism.  We  take  our  stand 
upon  the  facts  of  experience  and  estabHsh  upon  the 
systematized  formal  features  of  our  experience  a 
new  conception  of  the  a  priori,  recognizing  the  uni- 
versality and  necessity  of  formal  laws  but  rejecting 
Kant's  transcendental  idealism.  The  a  priori  is 
not  deducible  from  the  sensory  elements  of  our  sen- 
sations, but  we  trace  it  in  the  formal  features  of 
experience.  It  is  the  result  of  abstraction  and  sys- 
tematization.  Thus  we  establish  a  method  of  dealing 
with  experience  (commonly  called  Pure  Reason) 
which  is  possessed  of  universal  validity,  implying 
logical  necessity. 

The  New  Positivism  is  a  further  development  of 
philosophic  thought  which  combines  the  merits  of 
both  schools,  the  Transcendcntalists  and  Empiri- 
cists, in  a  higher  unity,  discarding  at  the  same  time 
their  al)crrations.  Tn  this  way  it  becomes  possible 
to  gain  a  firm  basis  upon  the  secure  ground  of  facts, 
according  to  the  principle  of  positivism,  and  yet  to 
preserve  a  method  esta])lished  by  a  study  of  the 
purely  formal,  wliich  will  not  end  in  nescience  (the 
ideal  of  agnosticism)  but  justify  science,  and  thus 
establishes  the  philosophy  of  science.' 


'  Wc  have  treated  the  philosophical  prnhlem  rif  the  a  priori  at 
full  length  in  a  discussion  of  Kant  s  Pmlcf^oinciia.  -See  the  author's 
Kant's  Prolegomena,  edited  in  English,  with  an  essay  on  Kant's  Phi- 
losophy and  other  Supplementary  Material  for  the  study  of  Kant, 
pp.   167-240.     Cf.  Fundamenlal  Problems,,  the  chapters  "Form  and 


40  THE  FOUNDATIONS  OF   MATHEMATICS. 

It  is  from  this  standpoint  of  the  philosophy  of 
science  that  we  propose  to  investigate  the  problem 
of  the  foundation  of  geometry. 

THE  A    PRIORI    AND  THE   PURELY    FORMAL. 

The  bulk  of  our  knowledge  is  from  experience, 
i.  e.,  we  know  things  after  having  become  acquainted 
with  them  .  Our  knowledge  of  things  is  a  posteriori. 
If  we  want  to  know  whether  sugar  is  sweet,  we  must 
taste  it.  If  we  had  not  done  so,  and  if  no  one  had 
tasted  it,  we  could  not  know  it.  However,  there  is 
another  kind  of  knowledge  which  we  do  not  find  out 
by  experience,  but  by  reflection.  If  I  want  to  know 
how  much  is  ^y^S^  or  (a-\-b)^  or  the  angles  in  a 
regular  polygon,  I  must  compute  the  answer  in 
my  own  mind.  I  need  make  no  experiments  but 
must  perform  the  calculation  in  my  own  thoughts. 
This  knowledge  which  is  the  result  of  pure  thought 
is  a  priori]  viz.,  it  is  generally  applicable  and  holds 
good  even  before  we  tried  it.  When  we  begin  to 
make  experiments,  we  presuppose  that  all  our  a 
priori  arguments,  logic,  arithmetic,  and  mathemat- 
ics, will  hold  good. 

Kant  declared  that  the  law  of  causation  is  of  the 
same  nature  as  arithmetical  and  logical  truths,  and 
that,  accordingly,  it  will  have  to  be  regarded  as 
a  priori.  Before  we  make  experiments,  we  know 
that  every  cause  has  its  effects,  and  wherever  there 

Formal  Thought,"  pp.  26-60,  and  "The  Old  and  the  New  Mathemat- 
ics," pp.  61-73;  and  Primer  of  Philosophy,  pp.  51-103. 


THE  PHILOSOPHICAL  BASIS.  4I 

is  an  effect  we  look  for  its  cause.  Causation  is  not 
proved  by,  but  justified  through,  experience. 

The  doctrine  of  the  a  priori  has  been  much  mis- 
interpreted, especially  in  England.  Kant  calls  that 
which  transcends  or  goes  beyond  experience  in  the 
sense  that  it  is  the  condition  of  experience  "tran- 
scendental," and  comes  to  the  conclusion  that  the 
a  priori  is  transcendental.  Our  a  priori  notions 
are  not  derived  from  experience  but  are  products 
of  pure  reflection  and  they  constitute  the  conditions 
of  experience.  By  experience  Kant  understands 
sense-impressions,  and  the  sense-impressions  of  the 
outer  world  (which  of  course  are  a  posteriori)  are 
reduced  to  system  by  our  transcendental  notions; 
and  thus  knowledge  is  the  product  of  the  a  priori 
and  the  a  posteriori. 

A  sense-impression  becomes  a  perception  by  be- 
ing regarded  as  the  effect  of  a  cause.  The  idea  of 
causation  is  a  transcendental  notion.  Without  it 
experience  would  be  impossible.  An  astronomer 
measures  angles  and  determines  the  distance  of  the 
moon  and  of  the  sun.  Experience  furnishes  the 
data,  they  are  a  posteriori;  but  his  mathematical 
methods,  the  number  system,  and  all  arithmetical 
functions  are  a  priori.  lie  knows  them  before  he 
collects  the  details  of  his  investigation:  and  in  so 
far  as  they  are  the  condition  without  which  his 
sense  -  impressions  could  not  be  transformed  into 
knowledge,  they  are  called  transcendental. 

Xote  here  Kant's  use  of  the  word  transcendental 
which  denotes  the  clearest  and  most  reliable  knowl- 


42  THE   FOUNDATIONS   OF   MATHEMATICS. 

edge  in  our  economy  of  thought,  pure  logic,  arith- 
metic, geometry,  etc.  But  transcendental  is  fre- 
quently (though  erroneously)  identified  with  "tran- 
scendent," which  denotes  that  which  transcends  our 
knowledge  and  accordingly  means  "unknowable." 
Whatever  is  transcendental  is,  in  Kantian  terminol- 
ogy, never  transcendent. 

That  much  w^ill  suffice  for  an  explanation  of  the 
historical  meaning  of  the  word  transcendental.  We 
must  now  explain  the  nature  of  the  a  priori  and  its 
source. 

The  a  priori  is  identical  with  the  purely  formal 
which  originates  in  our  mind  by  abstraction.  When 
we  limit  our  attention  to  the  purely  relational,  drop- 
ping all  other  features  out  of  sight,  we  produce  a 
field  of  abstraction  in  which  we  can  construct  purely 
formal  combinations,  such  as  numbers,  or  the  ideas 
of  types  and  species.  Thus  we  create  a  world  of 
pure  thought  which  has  the  advantage  of  being 
applicable  to  any  purely  formal  consideration  of 
conditions,  and  we  work  out  systems  of  numbers 
which,  when  counting,  we  can  use  as  standards  of 
reference  for  our  experiences  in  practical  life. 

But  if  the  sciences  of  pure  form  are  built  upon 
an  abstraction  from  which  all  concrete  features  are 
omitted,  are  they  not  empty  and  useless  verbiage? 

Empty  they  are,  that  is  true  enough,  but  for  all 
that  they  are  of  paramount  significance,  because 
they  introduce  us  into  the  sanctum  sanctissimum  of 
the  world,  the  intrinsic  necessity  of  relations,  and 
thus  they  become  the  key  to  all  the  riddles  of  the 


THE  PHILOSOPHICAL   BASIS.  43 

universe.  They  are  in  need  of  being  supplemented 
by  observation,  by  experience,  by  experiment ;  but 
while  the  mind  of  the  investigator  builds  up  purely 
formal  systems  of  reference  (such  as  numbers)  and 
]uirely  formal  space-relations  (such  as  geometry), 
the  essential  features  of  facts  (of  the  objective 
world)  are  in  their  turn,  too,  purely  formal,  and 
they  make  things  such  as  they  are.  The  suchness 
of  the  world  is  purely  formal,  and  its  suchness  alone 
is  of  importance. 

In  studying  the  processes  of  nature  we  watch 
transformations,  and  all  we  can  do  is  to  trace  the 
changes  of  form.  Matter  and  energy  are  words 
which  in  their  abstract  significance  have  little  value; 
they  merely  denote  actuality  in  general,  the  one  of 
being,  the  other  of  doing.  AMiat  interests  us  most 
are  the  forms  of  matter  and  energy,  how  they 
change  by  transformation;  and  it  is  obvious  that 
the  famous  law  of  the  conservation  of  matter  and 
energy  is  merely  the  reverse  of  the  truth  that  cau- 
sation is  transformation.  In  its  elements  which  in 
their  totality  are  called  matter  and  energy,  the  ele- 
ments of  existence  remain  the  same,  but  the  forms  in 
which  they  combine  change.  The  sum-total  of  the 
in.'i'^s  and  tlie  sum-total  of  the  forces  of  the  world 
can  be  neither  increased  nor  diminished;  thev  re- 
main the  same  to-day  that  they  ha\'e  ahvavs  been 
and  as  they  will  remain  forever. 

All  a  posteriori  cognition  is  concrete  and  par- 
ticular, while  all  a  priori  cognition  is  abstract  and 
general.     The  concrete  is  (at  least  in  its  relation 


44  THE   FOUNDATIONS  OF   MATHEMATICS. 

to  the  thinking-  subject)  incidental,  casual,  and  indi- 
vidual, but  the  abstract  is  universal  and  can  be  used 
as  a  general  rule  under  which  all  special  cases  may 
be  subsumed. 

The  a  priori  is  a  mental  construction,  or,  as  Kant 
says,  it  is  ideal,  viz.,  it  consists  of  the  stuff  that 
ideas  are  made  of,  it  is  mind-made.  While  we  grant 
that  the  purely  formal  is  ideal  we  insist  that  it  is 
made  in  the  domain  of  abstract  thought,  and  its 
fundamental  notions  ha\T  been  abstracted  from  ex- 
perience by  concentrating  our  attention  upon  the 
purely  formal.  It  is,  not  directl}^  but  indirectly  and 
ultimately,  derived  from  experience.  It  is  not  de- 
rived from  sense-experience  but  from  a  considera- 
tion of  the  relational  (the  purely  formal)  of  ex- 
perience. Thus  it  is  a  subjective  reconstruction  of 
certain  objective  features  of  experience  and  this 
reconstruction  is  made  in  such  a  way  as  to  drop 
every  thing  incidental  and  particular  and  retain 
only  the  general  and  essential  features ;  and  we  gain 
the  unspeakable  advantage  of  creating  rules  or  for- 
mulas which,  though  abstract  and  mind-made,  apply 
to  any  case  that  can  be  classified  in  the  same  cate- 
gory. 

Kant  made  the  mistake  of  identifying  the  term 
"ideal"  with  "subjective,"  and  thus  his  transcen- 
dental idealism  was  warped  by  the  conclusion  that 
our  purely  formal  laws  were  not  objective,  but  were 
imposed  by  our  mind  upon  the  objective  world.  Our 
mind  (Kant  said)  is  so  constituted  as  to  interpret 
all  facts  of  experience  in  terms  of  form,  as  appear- 


THE  PHILOSOPHICAL  BASIS.  45 

ing  in  space  and  time,  and  as  being  subject  to  the  law 
of  cause  and  eftect ;  but  what  things  are  in  them- 
selves w^e  cannot  know.  We  object  to  Kant's  sub- 
jectivizing  the  purely  formal  and  look  upon  form 
as  an  essential  and  inalienable  feature  of  objective 
existence.  The  thinking  subject  is  to  other  thinking 
subjects  an  object  moving  about  in  the  objective 
world  .  Even  v  hen  contemplating  our  own  exist- 
ence we  must  grant  (to  speak  with  Schopenhauer) 
that  our  bodily  actualization  is  our  own  object ;  i.  e., 
we  (each  one  of  us  as  a  real  living  creature)  are  as 
much  objects  as  are  all  the  other  objects  in  the 
world.  It  is  the  objectified  part  of  our  self  that  in 
its  inner  experience  abstracts  from  sense-experience 
the  interrelational  features  of  thino-s,  such  as  rioht 
and  left,  top  and  bottom,  shape  and  figure  and  struc- 
ture, succession,  connection,  etc.  The  formal  ad- 
heres to  the  object  and  not  to  the  subject,  and  every 
object  (as  soon  as  it  develops  in  the  natural  w^ay  of 
evolution  first  into  a  feeling  creature  and  then  into 
a  thinking  being)  will  be  abk^  to  build  up  a  priori 
from  the  abstract  notion  of  form  in  general  the  sev- 
eral systems  of  formal  thought :  arithmetic,  geom- 
etry, algebra,  logic,  and  the  conceptions  of  time, 
space,  and  causality. 

Accordingly,  all  formal  thought,  although  we 
grant  its  ideality,  is  fashioned  from  materials  ab- 
stracted from  the  ol)jecti\-e  world,  and  it  is  therefore 
a  matter  of  course  that  they  are  applicable  to  the 
objectixc  world.  They  belong  to  the  object  and, 
when  we  thinking  subjects  beget  them  t'rom  our  own 


46  THE   FOUNDATIONS  OF   MATHEMATICS. 

minds,  we  are  able  to  do  so  only  because  we  are  ob- 
jects that  live  and  move  and  have  our  being  in  the 
objective  world. 


AXYNESS  AND   ITS   UNIVERSALITY. 

We  know  that  facts  are  incidental  and  hap- 
hazard, and  appear  to  be  arbitrary ;  but  we  must  not 
rest  satisfied  with  single  incidents.  We  must  gather 
enough  single  cases  to  make  abstractions.  Abstrac- 
tions are  products  of  the  mind;  they  are  subjective; 
but  they  have  been  derived  from  experience,  and 
they  are  built  up  of  elements  that  have  objective 
significance. 

The  most  important  abstractions  ever  made  by 
man  are  those  that  are  purely  relational.  Every- 
thing from  which  the  sensory  element  is  entirely 
omitted,  where  the  material  is  disregarded,  is  called 
''pure  form,"  and  the  relational  being  a  considera- 
tion neither  of  matter  nor  of  force  or  energy,  but 
of  number,  of  position,  of  shape,  of  size,  of  form, 
of  relation,  is  called  "the  purely  formal."  The  no- 
tion of  the  purely  formal  has  been  gained  by  ab- 
straction, viz.,  by  abstracting,  i.  e.,  singling  out  and 
retaining,  the  formal,  and  by  thinking  away,  by  can- 
celling, by  omitting,  by  leaving  out,  all  the  features 
which  have  anything  to  do  with  the  concrete  sensory 
element  of  experience. 

And  what  is  the  result? 

We  retain  the  formal  element  alone  which  is 
void  of  all  concreteness,  void  of  all  materiality,  void 


THE  PHILOSOPHICAL  BASIS.  47 

of  all  particularity.  It  is  a  mere  nothing  and  a 
non-entity.  It  is  emptiness.  But  one  thing  is  left, 
— position  or  relation.  Actuality  is  replaced  by 
mere  potentiality,  viz.,  the  possible  conditions  of 
any  kind  of  being  that  is  possessed  of  form. 

The  word  "any"  denotes  a  simple  idea,  and  yet 
it  contains  a  good  deal  of  thought.  jMathematics 
builds  up  its  constructions  to  suit  any  condition. 
"Any"  implies  universality,  and  universality  in- 
cludes necessity  in  the  Kantian  sense  of  the  term. 

In  every  concrete  instance  of  an  experience  the 
subject-matter  is  the  main  thing  with  which  we  are 
concerned ;  but  the  purely  formal  aspect  is  after  all 
the  essential  feature,  because  form  determines  the 
character  of  things,  and  thus  the  formal  (on  account 
of  its  anyness)  is  the  key  to  their  comprehension. 

The  rise  of  man  above  the  animal  is  due  to  his 
ability  to  utilize  the  purely  formal,  as  it  revealed 
itself  to  him  especially  in  types  for  classifying  things, 
as  genera  and  species,  in  tracing  transformations 
which  present  themselves  as  effects  of  causes  and  re- 
ducing them  to  shapes  of  measurable  relations.  The 
ajjstraction  of  the  formal  is  made  through  the  in- 
strumentality of  language  and  ihc  result  is  reason, 
— the  faculty  of  abstract  thought.  Man  can  see  the 
universal  in  the  particular;  in  the  single  experiences 
he  can  trace  the  laws  that  are  generally  applicable 
to  cases  of  the  same  class;  he  observes  some  in- 
stances and  can  describe  tlu-ni  in  a  L^xiicral  formula  so 
as  to  cover  any  other  instance  of  the  same  kind,  and 
thus  he  becomes  master  of  the  situation;  he  learns 


48  THE   FOUNDATIONS  OF  MATHEMATICS. 

to  separate  in  thought  the  essential  from  the  acci- 
dental, and  so  instead  of  remaining  the  prey  of  cir- 
cumstance he  gains  the  power  to  adapt  circum- 
stances to  himself. 

Form  pervades  all  nature  as  an  essential  constit- 
uent thereof.  If  form  were  not  an  objective  feature 
of  the  world  in  w^hich  we  live,  formal  thought  would 
never  without  a  miracle,  or,  at  least,  not  without 
the  mvstery  of  mysticism,  have  originated  accord- 
ing to  natural  law,  and  man  could  never  have  arisen. 
But  form  being  an  objective  feature  of  all  existence, 
it  impresses  itself  in  such  a  way  upon  living  crea- 
tures that  rational  beings  will  naturally  develop 
among  animals  whose  organs  of  speech  are  per- 
fected as  soon  as  social  conditions  produce  that  de- 
mand for  communication  that  will  result  in  the  crea- 
tion of  language. 

The  marvelous  advantages  of  reason  dawned 
upon  man  like  a  revelation  from  on  high,  for  he  did 
not  invent  reason,  he  discovered  it;  and  the  senti- 
ment that  its  blessings  came  to  him  from  above, 
from  heaven,  from  that  power  which  sways  the  des- 
tiny of  the  whole  universe,  from  the  gods  or  from 
God,  is  as  natural  as  it  is  true.  The  anthropoid  did 
not  seek  reason :  reason  came  to  him  and  so  he  be- 
came man.  Man  became  man  by  the  grace  of  God, 
by  gradually  imbibing  the  Logos  that  was  with  God 
in  the  beginning ;  and  in  the  dawn  of  human  evolu- 
tion we  can  plainly  see  the  landmark  of  mathemat- 
ics, for  the  first  grand  step  in  the  development  of 


THE  PHILOSOPHICAL  BASIS.  49 

man  as  distinguished  from  the  transitional  forms  of 
the  anthropoid  is  the  abihty  to  count. 

Man's  distinctive  characteristic  remains,  even 
to-day,  reason,  the  f acuhy  of  purely  formal  thought ; 
and  the  characteristic  of  reason  is  its  general  appli- 
cation. All  its  verdicts  are  universal  and  involve 
apriority  or  beforehand  knowledge  so  that  man  can 
foresee  events  and  adapt  means  to  ends. 

APRIORITY   OF   DIFFERENT   DEGREES. 

Kant  has  pointed  out  the  kinship  of  all  purely 
formal  notions.  The  validity  of  mathematics  and 
logic  assures  us  of  the  validity  of  the  categories 
including  the  conception  of  causation;  and  yet  ge- 
ometry cannot  be  derived  from  i)ure  reason  alone, 
but  contains  an  additional  element  which  imparts 
to  its  fundamental  conceptions  an  arbitrary  appear- 
ance if  we  attempt  to  treat  its  deductions  as  rigidly 
a  priori.  Why  should  there  be  straight  lines  at  all  ? 
Why  is  it  possible  that  by  quartering  the  circle  we 
should  have  right  angles  with  all  their  peculiarities? 
All  these  and  similar  notions  can  not  be  subsumed 
under  a  general  formula  of  pure  reason  from  which 
we  could  derive  it  with  logical  necessity. 

When  dealing  with  lines  we  observe  their  exten- 
tion  in  one  direction,  when  dealing  with  jilanes  we 
have  two  dimensions,  when  measuring  solids  we 
have  three.  Why  can  we  not  continue  and  construct 
bodies  that  extend  in  four  (h'niensions?  The  limit 
set  us  by  space  as  it  posit i\cl\-  presents  itself  to  us 


50  THE  FOUNDATIONS  OF   MATHEMATICS. 

seems  arbitrary,  and  while  transcendental  truths 
are  undeniable  and  obvious,  the  fundamental  no- 
tions of  geometry  seem  as  stubborn  as  the  facts  of 
our  concrete  existence.  Space,  generally  granted 
to  be  elbow-room  for  motion  in  all  directions,  after 
all  appears  to  be  a  definite  magnitude  as  much  as 
a  stone  wall  which  shuts  us  in  like  a  prison,  allowing 
us  to  proceed  in  such  a  way  only  as  is  permissible 
by  those  co-ordinates  and  no  more.  We  can  by 
no  resort  break  through  this  limitation.  Verily  we 
might  more  easily  shatter  a  rock  that  impedes  our 
progress  than  break  into  the  fourth  dimension.  The 
boundary  line  is  inexorable  in  its  adamantine  rigor. 

Considering  all  these  unquestionable  statements, 
is  there  not  a  great  probability  that  space  is  a  con- 
crete fact  as  positive  as  the  existence  of  material 
things,  and  not  a  mere  form,  not  a  mere  potentiality 
of  a  general  nature  ?  Certainly  Euclidean  geometry 
contains  some  such  arbitrary  elements  as  we  should 
expect  to  meet  in  the  realm  of  the  a  posteriori.  No 
wonder  that  Gauss  expressed  "the  desire  that  the 
Euclidean  geometry  should  not  be  the  true  geom- 
etry," because  "in  that  event  we  should  have  an  ab- 
solute measure  a  priori." 

Are  we  thus  driven  to  the  conclusion  that  our 
space-conception  is  not  a  priori:  and  if,  indeed,  it  is 
not  a  priori,  it  must  be  a  posferioril  What  else  can 
it  be?     Terfiiim  non  dafur. 

If  we  enter  more  deeply  into  the  nature  of  the 
a  priori,  we  shall  learn  that  there  are  different  kinds 


THE  PHILOSOPHICAL  BASIS.  5 1 

of  apriority,  and  there  is  a  difference  between  the 
logical  a  priori  and  the  geometrical  a  priori. 

Kant  never  investigated  the  source  of  the  a  pri- 
ori. He  discovered  it  in  the  mind  and  seemed  satis- 
fied with  the  notion  that  it  is  the  nature  of  the  mind 
to  be  possessed  of  time  and  space  and  the  categories. 
He  went  no  further.  He  never  asked,  how  did  mind 
originate? 

Had  Kant  inquired  into  the  origin  of  mind,  he 
would  have  found  that  the  a  priori  is  woven  into 
the  texture  of  mind  by  the  uniformities  of  experi- 
ence. The  uniformities  of  experience  teach  us  the 
laws  of  form,  and  the  purely  formal  applies  not  to 
one  case  only  but  to  any  case  of  the  same  kind,  and 
so  it  involves  "anyness,"  that  is  to  say,  it  is  a  priori. 

Mind  is  the  ])roduct  of  memory,  and  we  may 
briefly  describe  its  origin  as  follows : 

Contact  with  the  outer  world  produces  impres- 
sions in  sentient  substance.  The  traces  of  these 
impressions  are  preserved  (a  condition  \\hich  is 
called  "memory")  and  they  can  be  re\i\ed  (  w hicli 
state  is  called  "recollection").  Sense-impressions 
are  different  in  kind  and  leave  dift'ercnt  traces,  but 
those  which  are  the  same  in  kind,  or  similar,  leave 
traces  the  forms  of  which  arc  the  same  or  similar; 
and  sense-impressions  of  the  same  kind  arc  regis- 
tered in  the  traces  having  the  same  form.  As  a  note 
of  a  definite  pitch  makes  chords  of  the  same  pitch 
vibrate  while  it  passes  all  others  by;  so  new  sense- 
imj)rc^^ions  rc\'ive  those  tracer  f»n1\-  into  \\liicli  tliev 
fit.  nnd  t]i('r{'b\-  ,'innoiinre  llicnisclN'es  as  bciuLZ'  the 


52  THE  FOUNDATIONS  OF   MATHEMATICS. 

same  in  kind.  Thus  all  sense-impressions  are  sys- 
tematized according  to  their  forms,  and  the  result 
is  an  orderly  arrangement  of  memories  which  is 
called  "mind."^ 

Thus  mind  develops  through  uniformities  in  sen- 
sation according  to  the  laws  of  form.  Whenever 
a  new  sense-perception  registers  itself  mechanically 
and  automatically  in  the  trace  to  which  it  belongs, 
the  event  is  tantamount  to  a  logical  judgment  which 
declares  that  the  object  represented  by  the  sense- 
impression  belongs  to  the  same  class  of  objects 
which  produced  the  memory  traces  with  which  it  is 
registered. 

If  we  abstract  the  interrelation  of  all  memory- 
traces,  omitting  their  contents,  we  have  a  pure  sys- 
tem of  genera  and  species,  or  the  a  priori  idea  of 
"classes  and  subclasses." 

The  a  priori,  though  mind-made,  is  constructed 
of  chips  taken  from  the  objective  world,  but  our 
several  a  priori  notions  are  by  no  means  of  one  and 
the  same  nature  and  rigidity.  On  the  contrary, 
there  are  different  degrees  of  apriority.  The  emp- 
tiest forms  of  pure  thought  are  the  categories,  and 
the  most  rigid  truths  are  the  logical  theorems,  which 
can  be  represented  diagrammatically  so  as  to  be  a  de- 
uwnstratio  ad  ociilos. 

If  all  bs  are  B  and  if  /S  is  a  5,  then  yS  is  a  B.  If 
all  dogs  are  quadrupeds  and  if  all  terriers  are  dogs, 
then  terriers  are  quadrupeds.    It  is  the  most  rigid 

^  For  a  more  detailed  exposition  see  the  author's  Soul  of  Man; 
also  his  Whence  and  Whither. 


THE  PHILOSOPHICAL   BASIS.  53 

kind  of  argument,  and  its  statements  are  practically 
tautologies. 

The  case  is  different  with  causation.  The  class 
of  abstract  notions  of  which  causation  is  an  instance 
is  much  more  complicated.  No  one  doubts  that 
every  effect  must  have  had  its  cause,  but  one  of  the 
keenest  thinkers  was  in  deep  earnest  when  he 
doubted  the  possibility  of  proving  this  obvious  state- 
ment. And  Kant,  seeing  its  kinship  with  geometrv 
and  algebra,  accepted  it  as  a  priori  and  treated  it 
as  being  on  equal  terms  with  mathematical  axioms. 
Yet  there  is  an  additional  element  in  the  formula 
of  causation  which  somehow  disguises  its  a  priori 
origin,  and  the  reason  is  that  it  is  not  as  rigidly 
a  priori  as  are  the  norms  of  pure  logic. 

What  is  this  additional  element  that  somehow 
savors  of  the  a  posteriori? 

If  we  contemplate  the  interrelation  of  genera 
and  species  and  su1)species,  we  find  that  the  cate- 
gories with  which  we  operate  are  at  rest.  They 
stand  before  us  like  a  ^\cll  arranged  cabinet  with 
several  di\-isions  and  drawers,  and  these  drawers 
have  subdivisions  and  in  these  subdivisions  we  keep 
1)0xes.  The  cabinet  is  our  (/  priori  system  of  classi- 
fication and  we  store  in  it  our  a  posteriori  ini])res- 
sions.  If  a  thing  is  in  box  /8,  we  seek  for  it  in 
drawer  b  which  is  a  subdivision  of  the  dc])artiiienl  !>. 

How  different  is  causation!  W'liile  in  logic 
everything  is  at  rest,  causation  is  not  conccixablc 
without  motion.  The  norms  f)f  j)urc'  reason  are 
static,  the  law  of  cause  and  effect  is  (Knaniic;  and 


54  THE   FOUNDATIONS  OF   MATHEMATICS. 

thus  we  have  in  the  conception  of  cause  and  effect 
an  additional  element  which  is  mobility. 

Causation  is  the  law  of  transformation.  We  have 
a  definite  system  of  interrelated  items  in  which  we 
observe  a  change  of  place.  The  original  situation 
and  all  detailed  circumstances  are  the  conditions; 
the  motion  that  produces  the  change  is  the  cause; 
the  result  or  new  arrangement  of  the  parts  of  the 
whole  system  is  the  effect.  Thus  it  appears  that  cau- 
sation is  only  another  version  of  the  law  of  the  con- 
servation of  matter  and  energy.  The  concrete  items 
of  the  whole  remain,  in  their  constitutional  elements, 
the  same.  No  energy  is  lost ;  no  particle  of  matter 
is  annihilated;  and  the  change  that  takes  place  is 
mere  transformation.^ 

The  law  of  causation  is  otherwise  in  the  same 
predicament  as  the  norms  of  logic.  It  can  never  be 
satisfactorily  proved  by  experience.  Experience 
justifies  the  a  priori  and  verifies  its  tenets  in  single 
instances  which  prove  true,  but  single  instances  can 
never  demonstrate  the  universal  and  necessary  va- 
lidity of  any  a  priori  statement. 

The  logical  a  priori  is  rigidly  a  priori:  it  is  the 
a  priori  of  pure  reason.  But  there  is  another  kind 
of  a  priori  which  admits  the  use  of  that  other  ab- 
stract notion,  mobility,  and  mobility  as  much  as 
form  is  part  and  parcel  of  the  thinking  mind.  Our 
conception  of  cause  and  efifect  is  just  as  ideal  as  our 
conception  of  genera  and  species.    It  is  just  as  much 

'See  the  author's  Ursachc,  Gnind  iiin!  Zzvcck.  Dresden,  1883; 
also  his  Fundamental  Problems. 


THE  PHILOSOPHICAL   BASIS.  55 

mind-made  as  they  are,  and  its  intrinsic  necessity 
and  universal  validity  are  the  same.  Its  apriority 
cannot  be  doubted ;  but  it  is  not  rigidly  a  priori,  and 
we  will  call  it  purely  a  priori. 

We  may  classify  all  a  priori  notions  under  two 
headings  and  both  ar-^  transcendental  (viz.,  con- 
ditions of  knowledge  in  their  special  fields) :  one  is 
the  a  priori  of  being,  the  other  of  doing.  The  rigid 
a  priori  is  passive  anyness,  the  less  rigid  a  prion 
is  active  anyness.  Geometry  belongs  to  the  latter. 
Its  fundamental  concept  of  space  is  a  product  of  ac- 
tive apriority;  and  thus  we  cannot  derive  its  laws 
from  pure  logic  alone. 

The  main  difficulty  of  the  parallel  theorem  and 
the  straight  line  consists  in  our  space-conception 
which  is  not  derived  from  rationality  in  general, 
but  results  from  our  contemplation  of  motion.  Our 
space-conception  accordingly  is  not  an  idea  of  pure 
reason,  but  the  product  of  pure  activity. 

Kant  felt  the  difference  and  distinguished  be- 
tween pure  reason  and  pure  intuition  or  Anschau- 
nng.  ?Ie  did  not  expressly  say  so,  but  his  treatment 
suggests  the  idea  that  we  ought  to  distinguish  be- 
tween two  different  kinds  of  a  priori.  Transcen- 
dental logic,  and  with  it  all  common  notions  of 
Euclid,  arc  mere  applications  of  the  law  of  consist- 
ency; they  are  "rigidly  a  priori.''  P>ut  our  pure 
space-conception  presupposes,  in  addition  to  pure 
reason,  our  own  activity,  the  potentiality  of  moving 
about  in  anv  kind  of  a  field,  and  thus  it  admits 
another  factor  which  c.-mnol  be  derived  from  pure 


56  THE  FOUNDATIONS  OF   MATHEMATICS. 

reason  alone.  Hence  all  attempts  at  proving  the 
theorem  on  rigidly  a  priori  grounds  have  proved 
failures. 

SPACE  AS  A  SPREAD  OF  MOTION. 

Mathematicians  mean  to  start  from  nothingness, 
so  they  think  away  everything,  but  they  retain  their 
own  mentality.  Though  even  their  mind  is  stripped 
of  all  particular  notions,  they  retain  their  principles 
of  reasoning  and  the  privilege  of  moving  about, 
and  from  these  two  sources  geometry  can  be  con- 
structed. 

The  idea  of  causation  goes  one  step  further:  it 
admits  the  notions  of  matter  and  energy,  emptied 
of  all  particularity,  in  their  form  of  pure'generali- 
zations.  It  is  still  a  priori,  but  considerably  more 
complicated  than  pure  reason. 

The  field  in  which  the  geometrician  starts  is  pure 
nothingness ;  but  we  shall  learn  later  on  that  noth- 
ingness is  possessed  of  positive  qualifications.  We 
must  therefore  be  on  our  guard,  and  we  had  better 
inquire  into  the  nature  and  origin  of  our  nothing- 
ness. 

The  geometrician  cancels  in  thought  all  positive 
existence  except  his  own  mental  activity  and  starts 
moving  about  as  a  mere  nothing.  In  other  words, 
we  establish  by  abstraction  a  domain  of  monotonous 
sameness,  which  possesses  the  advantage  of  "any- 
ness,"  i.  e.,  an  absence  of  particularity  involving 
universal  validity.  In  this  field  of  motion  we  pro- 
ceed to  produce  geometrical  constructions. 


THE  PHILOSOPHICAL   BASIS.  ^J 

The  geometrician's  activity  is  pure  motion, 
which  means  that  it  is  mere  progression ;  the  ideas 
of  a  force  exerted  in  moving  and  also  of  resistance 
to  be  overcome  are  absolutely  excluded. 

We  start  moving,  but  whither?  Before  us  are 
infinite  possibilities  of  direction.  The  inexhaus- 
tibility of  chances  is  part  of  the  indifference  as 
to  definiteness  of  determining  the  mode  of  motion 
(be  it  straight  or  curved).  Let  us  start  at  once 
in  all  possible  directions  which  are  infinite,  (a  propo- 
sition which,  in  a  way,  is  realized  by  the  light), 
and  having  proceeded  an  infinitesimal  way  from  the 
starting-point  A  to  the  points  B.  Bi,,  B2,  B3,  B4, 
.  .  .  .  B^  ;  we  continue  to  move  in  infinite  directions 
at  each  of  these  stations,  reaching  from  B  the  points 
C,  Ci,  C2,  C3,  C4, .  .  .  .  C^.  From  Bi  we  would  switch 
ofT  to  the  points  QW  Cf ,  Cf ^  Cf , ....  C^^  etc.  until 
we  reach  from  B^  the  points  C^",  Cf*,  Cf»,  Cf*, 
•  •  •  -C^",  thus  exhausting  all  the  points  which  clus- 
ter around  every  Bi,  Bo,  B..,  Bj B„.     Thus,  Ijy 

moving  after  the  fashion  of  the  light,  spreading 
again  and  again  from  each  new  ])oint  in  all  direc- 
tions, in  a  medium  that  offers  no  resistance  what- 
e\cr,  VsC  ol)tain  a  uniform  spread  of  light  whose 
intensity  in  every  ])oint  is  in  the  inverse  square  of 
its  distance  from  its  source.  Every  lighted  spot  ])e- 
comes  a  center  of  its  own  from  which  light  travels 
on  in  all  directions.  But  among  these  infinite  direc- 
tions there  are  rays,  A,  B,  C, Ai.  Bi,  Ci, 

A2,  B2.  C2 etc.,  that  is  to  say,  lines  of  motion 

that  pursue  the  original  direction  and  arc  paths  of 


58  THE   FOUNDATIONS   OF   MATHEMATICS. 

maximum  intensity.  Each  of  these  rays,  thus  ide- 
ally constructed,  is  a  representation  of  the  straight 
line  which  being  the  shortest  path  between  the  start- 
ing-point A  and  any  other  point,  is  the  climax  of 
directness :  it  is  the  upper  limit  of  effectiveness  and 
its  final  boundary,  a  noii  plus  ultra.  It  is  a  maxi- 
mum because  there  is  no  loss  of  efficacy.  The 
straight  line  represents  a  climax  of  economy,  viz., 
the  greatest  intensity  on  the  shortest  path  that  is 
reached  among  infinite  possibilities  of  progression 
by  uniformly  following  up  all.  In  every  ray  the 
maximum  of  intensity  is  attained  by  a  minimum 
of  progression. 

Our  construction  of  motion  in  all  directions  after 
the  fashion  of  light  is  practically  pure  space;  but 
to  avoid  the  forestalling  of  further  implications  we 
will  call  it  simply  the  spread  of  motion  in  all  direc- 
tions. 

The  path  of  highest  intensity  in  a  spread  of 
motion  in  all  directions  corresponds  to  the  ray  in  an 
ideal  conception  of  a  spread  of  light,  and  it  is  equiv- 
alent to  the  straight  line  in  geometry. 

We  purposely  modify  our  reference  to  light  in 
our  construction  of  straight  lines,  for  we  are  well 
aware  of  the  fact  that  the  notion  of  a  ray  of  light 
as  a  straight  line  is  an  ideal  which  describes  the 
progression  of  light  only  as  it  appears,  not  as  it  is. 
The  physicist  represents  light  as  rays  only  when 
measuring  its  effects  in  reflection,  etc.,  but  when 
considering  the  nature  of  light,  he  looks  upon  rays 
as  transversal  oscillations  of  the  ether.    The  notion 


THE  PHILOSOPHICAL   BASIS.  59 

of  light  as  rays  is  at  bottom  as  much  an  a  priori 
construction  as  is  Newton's  formula  of  gravitation. 

The  construction  of  space  as  a  spread  of  motion 
in  all  directions  after  the  analogy  of  light  is  a  sum- 
mary creation  of  the  scope  of  motion,  and  we  call  it 
"ideal  space."  Everything  that  moves  about,  if  it 
develops  into  a  thinking  subject,  when  it  forms  the 
abstract  idea  of  mobility,  \\\\\  inevitably  create  out 
of  the  data  of  its  own  existence  the  ideal  "scope  of 
motion,"  which  is  space. 

AMien  the  geometrician  starts  to  construct  his 
figures,  drawing  lines  and  determining  the  position 
of  points,  etc.,  he  tacitly  presupposes  the  existence 
of  a  spread  of  motion,  such  as  we  have  described. 
Motility  is  part  of  his  equipment,  and  motility  pre- 
supposes a  field  of  motion,  viz.,  space. 

Space  is  the  possibility  of  motion,  and  by  ideally 
moving  about  in  all  possible  directions  the  number 
of  which  is  inexliaustil^le,  we  construct  our  notion 
of  inu'c  space.  If  we  speak  of  space  we  mean  this 
construction  of  our  mobility.  It  is  an  a  priori  con- 
struction and  is  as  unique  as  logic  or  arithmetic. 
There  is  but  one  space,  and  all  spaces  arc  but  por- 
tions of  this  one  construction.  The  problem  of  tri- 
dimcnsionality  will  1)c  considered  later  on.  Here 
we  insist  only  on  the  objective  validity  of  our  a 
priori  construction,  w'hich  is  the  same  as  the  ob- 
jective validity  of  all  our  a  priori  constructions — 
of  logic  and  aritbmetic  and  causality,  and  it  rests 
ti])on  the  same  foundation.  Our  mathematical  .space 
omits  all  particulriritv  and   serves  our  purpose  of 


60  THE   FOUNDATIONS  OF   MATHEMATICS. 

universal  application:  it  is  founded  on  "anyness," 
and  thus,  within  the  limits  of  its  abstraction,  it  holds 
good  everywhere  and  under  all  conditions. 

There  is  no  need  to  find  out  by  experience  in  the 
domain  of  the  a  posteriori  whether  pure  space  is 
curved.  Anyness  has  no  particular  qualities;  we 
create  this  anyness  by  abstraction,  and  it  is  a  matter 
of  course  that  in  the  field  of  our  abstraction,  space 
will  be  the  same  throughout,  unless  by  another  act 
of  our  creative  imagination  we  appropriate  partic- 
ular qualities  to  different  regions  of  space. 

The  fabric  of  which  the  purely  formal  is  woven 
is  an  absence  of  concreteness.  It  is  (so  far  as  mat- 
ter is  concerned)  nothing.  Yet  this  airy  nothing 
is  a  pretty  tough  material,  just  on  account  of  its 
indifferent  "any"-ness.  Being  void  of  particularity, 
it  is  universal ;  it  is  the  same  throughout,  and  if  we 
proceed  to  build  our  air-castles  in  the  domain  of 
anyness,  we  shall  find  that  considering  the  absence 
of  all  particularity  the  same  construction  will  be  the 
same,  wherever  and  whenever  it  may  be  conceived. 

Professor  Clifford  says  :^  ''We  assume  that  two 
lengths  which  are  equal  to  the  same  length  are 
equal  to  each  other."  But  there  is  no  "assumption" 
about  it.  The  atmosphere  in  which  our  mathemat- 
ical creations  are  begotten  is  sameness.  Therefore 
the  same  construction  is  the  same  wherever  and 
whenever  it  may  be  made.  We  consider  form  only ; 
we  think  away  all  other  concrete  properties,  both 

*Loc.  cit.,  p.  53. 


THE   PHILOSOPHICAL   BASIS.  6l 

of  matter  and  energ-y,  mass,  weight,  intensity,  and 
qualities  of  any  kind. 

UNIQUEXESS  OF  PURE  SPACE. 

Our  thought-forms,  constructed  in  the  reahii  of 
empty  abstraction,  serve  as  models  or  as  systems  of 
reference  for  any  of  our  observations  in  the  real 
world  of  sense-experience.  The  laws  of  form  are 
as  well  illustrated  in  our  models  as  in  real  things, 
and  can  be  derived  from  either ;  but  the  models  of 
our  thought-forms  are  always  ready  at  hand  while 
the  real  things  are  mostly  inaccessible.  The  any- 
ness  of  pure  form  explains  the  parallelism  that  ob- 
tains between  our  models  and  actual  experience, 
which  was  puzzling  to  Kant.  And  truly  at  first 
sight  it  is  mystifying  that  a  pure  thought-construc- 
tion can  reveal  to  us  some  of  the  most  im])orlant 
and  deepest  secrets  of  objective  nature;  but  the  sim- 
])le  solution  of  the  mystery  consists  in  this,  that  the 
actions  of  nature  are  determined  by  the  same  con- 
ditions of  possil)le  motions  with  which  pure  thought 
is  confronted  in  its  efforts  to  construct  its  models. 
1 1  ere  as  well  as  there  we  have  consistencv,  thai  is 
to  say,  a  thing  done  is  uTii(|Ucly  determined,  and, 
in  pure  thought  as  well  as  in  reality,  it  is  such  as  it 
has  been  made  by  construction  . 

Our  constructions  are  made  in  anvncss  and  ai)ply 
to  all  possible  instances  of  the  kind;  and  thus  we 
may  as  well  define  space  as  the  potentiality  of  meas- 
uring, which  presupposes  moving  about.     Mobility 


62  THE   FOUNDATIONS   OF    MATIIEMx\TICS. 

granted,  we  can  construct  space  as  the  scope  of  our 
motion  in  anyness.  Of  course  we  must  bear  in  mind 
that  our  motion  is  in  thou^'ht  only  and  we  have 
dropped  all  notions  of  particularity  so  as  to  leave 
an  utter  absence  of  force  and  resistance.  The  motor 
element,  qua  energy,  is  not  taken  into  consideration, 
but  we  contemplate  only  the  products  of  progres- 
sion. 

Since  in  the  realm  of  pure  form,  thus  created 
by  abstraction,  we  move  in  a  domain  void  of  par- 
ticularity, it  is  not  an  assumption  (as  Riemann 
declares  in  his  famous  inaugural  dissertation),  but 
a  matter  of  course  which  follows  with  logical  ne- 
cessity, that  lines  are  independent  of  position;  they 
are  the  same  anywhere. 

In  actual  space,  position  is  by  no  means  a  negli- 
gible quantity.  A  real  pyramid  consisting  of  actual 
material  is  possessed  of  different  qualities  according 
to  position,  and  the  line  AB,  representing  a  path 
from  the  top  of  a  mountain  to  the  valley  is  very 
different  from  the  line  BA,  which  is  the  path  from 
the  valley  to  the  top  of  the  mountain.  In  Euclidean 
geometry  AB=BA. 

Riemann  attempts  to  identify  the  mathematical 
space  of  a  triple  manifold  wath  actual  space  and  ex- 
pects a  proof  from  experience,  but,  properly  speak- 
ing, they  are  radically  different.  In  real  space  po- 
sition is  not  a  negligible  factor,  and  would  necessi- 
tate a  fourth  co-ordinate  which  has  a  definite  rela- 
tion to  the  plumb-line;  and  this  fourth  co-ordinate 
(which  we  may  call  a  fourth  dimension)  suffers  a 


THE  PHILOSOPHICAL  BASIS.  63 

constant  modification  of  increase  in  inverse  propor- 
tion to  the  square  of  the  distance  from  the  center  of 
this  planet  of  ours.  It  is  rectihnear,  yet  all  the  plumb- 
lines  are  converging  toward  an  inaccessible  center; 
accordingly,  they  are  by  no  means  of  equal  value  in 
their  different  parts.  How  different  is  mathemat- 
ical space !  It  is  homogeneous  throughout.  And  it 
is  so  because  we  made  it  so  by  abstraction. 

Pure  form  is  a  feature  which  is  by  no  means 
a  mere  nonentity.  Having  emptied  existence  of  all 
concrete  actuality,  and  having  thought  away  every- 
thing, we  are  confronted  by  an  absolute  vacancy — 
a  zero  of  existence :  but  the  zero  has  positive  char- 
acteristics and  there  is  this  peculiarity  about  the 
zero  that  it  is  the  mother  of  infinitude.  The  thought 
is  so  true  in  mathematics  that  it  is  trite.  Let  any 
number  be  divided  by  nought,  the  result  is  the  in- 
fmitely  great ;  and  let  nought  be  divided  by  any 
number,  the  result  is  the  infinitely  small.  In  think- 
ing away  everything  concrete  we  retain  with  our 
nothingness  potentiality.  Potentiality  is  the  empire 
of  purely  formal  constructions,  in  the  dim  back- 
ground of  which  lurks  the  phantom  of  infinitude. 

MATHEMATICAL   SPACi:   AND    f'l  I YSIOLOGICAL   SPACE. 

If  we  admit  lo  our  conccplion  of  s])ace  the  (|ual- 
iiics  f)i  bodies  such  as  mass,  our  conception  of  real 
space  will  become  more  comj)licated  still.  \\'liai 
we  gain  in  concrete  definiteness  we  lose  from  uni- 
versality, and  we  can  return  to  the  general  appli- 


64  THE   FOUNDATIONS  OF  MATHEMATICS. 

cability  of  a  priori  conditions  only  by  dropping 
all  concrete  features  and  limiting  our  geometrical 
constructions  to  the  abstract  domain  of  pure  form. 

Mathematical  space  with  its  straight  lines, 
planes,  and  right  angles  is  an  ideal  construction. 
It  exists  in  our  mind  only  just  as  much  as  do  logic 
and  arithmetic.  In  the  external  world  there  are 
no  numbers,  no  mathematical  lines,  no  logarithms, 
no  sines,  tangents,  nor  secants.  The  same  is  true 
of  all  the  formal  sciences.  There  are  no  genera  and 
species,  no  syllogisms,  neither  inductions  nor  deduc- 
tions, running  about  in  the  world,  but  only  concrete 
individuals  and  a  concatenation  of  events.  There 
are  no  laws  that  govern  the  motions  of  stars  or  mol- 
ecules ;  yet  there  are  things  acting  in  a  definite  way, 
and  their  actions  depend  on  changes  in  relational 
conditions  which  can  be  expressed  in  formulas.  All 
the  generalized  notions  of  the  formal  sciences  are 
mental  contrivances  which  comprise  relational  fea- 
tures in  general  rules.  The  formulas  as  such  are 
purely  ideal,  but  the  relational  features  which  they 
describe  are  o1)jectively  real. 

Thus,  the  space-conception  of  the  mathematician 
is  an  ideal  construction ;  but  the  ideal  has  objective 
significance.  Ideal  and  subjective  are  by  no  means 
synonyms.  With  the  help  of  an  ideal  space-con- 
ception we  can  acquire  knowledge  concerning  the 
real  space  of  the  objective  world.  Here  the  New- 
tonian law  may  be  cited  as  a  conspicuous  example. 

How  can  the  thinking  subject  know  a  priori  any- 
thing about  the  object?    Simply  because  the  subject 


THE  PHILOSOPHICAL  BASIS.  05 

is  an  object  moving  about  among  other  objects. 
Mobility  is  a  qualification  of  the  object,  and  I,  the 
thinking  subject,  become  conscious  of  the  general 
rules  of  motion  only,  because  I  also  am  an  object 
endowed  with  mobility.  My  "scope  of  motion"  can- 
not be  derived  from  the  abstract  idea  of  myself  as 
a  thinking  subject,  but  is  the  product  of  a  conside- 
ration of  my  mobility,  generalized  from  my  activities 
as  an  object  by  omitting  all  particularities. 

Mathematical  space  is  a  priori  in  the  Kantian 
sense.  However,  it  is  not  ready-made  in  our  mind, 
it  is  not  an  innate  idea,  but  the  product  of  much 
toil  and  careful  thought.  Nor  will  its  construction 
be  possible,  except  at  a  maturer  age  after  a  long 
development. 

Physiological  space  is  the  direct  and  unsophisti- 
cated space-conception  of  our  senses.  It  originates 
through  experience,  and  is,  in  its  way,  a  truer  pic- 
ture of  actual  or  physical  space  than  mathematical 
space.  The  latter  is  more  general,  the  former  more 
concrete.  Tn  ])liysiological  space  position  is  not  in- 
different, for  high  and  low,  right  and  left,  and  up 
and  down  are  of  great  importance.  Geometrically 
congruent  figures  ])roduce  (as  Mach  has  shown) 
remarkably  different  im])ressions  if  they  present 
themselves  to  the  eyes  in  different  positions. 

Tn  a  geometrical  plane  the  figures  can  be  shoved 
about  without  suffering  a  change  of  form.  If  they 
are  flopped,  tlicir  inner  relations  remain  the  same, 
as,  e.i^.,  helices  of  opposite  directions  are,  mathe- 
maticallv  considered,  cmiifrucnt.  while  in  actual  life 


66  THE   FOUNDATIONS  OF   MATHEMATICS. 

they  would  always  remain  mere  symmetrical  coun- 
terparts. So  the  right  and  the  left  hands  considered 
as  mere  mathematical  bodies  are  congruent,  while 
in  reality  neither  can  take  the  place  of  the  other. 
A  glove  which  we  may  treat  as  a  two-dimensional 
thing  can  be  turned  inside  out,  but  we  would  need 
a  fourth  dimension  to  flop  the  hand,  a  three-dimen- 
sional body,  into  its  inverted  counterpart.  So  long- 
as  we  have  no  fourth  dimension,  the  latter  being  a 
mere  logical  fiction,  this  cannot  be  done.  Yet  mathe- 
matically considered,  the  two  hands  are  congruent. 
Why?  Not  because  they  are  actually  of  the  same 
shape,  but  because  in  our  mathematics  the  quali- 
fications of  position  are  excluded ;  the  relational 
alone  counts,  and  the  relational  is  the  same  in  both 
cases. 

Mathematical  space  being  an  ideal  construction, 
it  is  a  matter  of  course  that  all  mathematical  prob- 
lems must  be  settled  by  a  priori  operations  of  pure 
thought,  and  cannot  be  decided  by  external  experi- 
ment or  by  reference  to  a  posteriori  information. 

HOMOGENEITY  OF  SPACE  DUE  TO  ABSTRACTION. 

\Mien  moving  about,  we  change  our  place  and 
pass  by  different  objects.  These  objects  too  are 
moving;  and  thus  our  scope  of  motion  tallies  so 
exactly  with  theirs  that  one  can  be  used  for  the  com- 
putation of  the  other.  All  scopes  of  motion  are  pos- 
sessed of  the  same  anyness. 

Space  as  we  find  it  in  experience  is  best  defined 


THE  PHILOSOPHICAL   BASIS.  67 

as  the  juxtaposition  of  things.  If  there  is  need  of 
distinguishing  it  from  our  ideal  space-conception 
which  is  the  scope  of  our  mobihty,  we  may  call  the 
former  pure  objective  space,  the  latter  pure  sub- 
jective space,  but,  our  subjective  ideas  being  rooted 
in  our  mobility,  which  is  a  constitutional  feature 
of  our  objective  existence,  for  many  practical  pur- 
poses the  two  are  the  same. 

But  though  pure  space,  whether  its  conception 
be  established  objectively  or  subjectively,  must  be 
accepted  as  the  same,  are  we  not  driven  to  the  con- 
clusion that  there  are  after  all  two  different  kinds 
of  space:  mathematical  space,  which  is  ideal,  and 
physiological  space,  which  is  real?  And  if  they  arc 
different,  must  we  not  assume  them  to  be  indepen- 
dent of  each  other  ?    \Miat  is  their  mutual  relation  ? 

The  two  spaces,  the  ideal  construction  of  mathe- 
matical space  and  the  reconstruction  in  our  senses 
of  the  juxtaposition  of  things  surrounding  us,  are 
different  solely  because  they  have  been  l)ui]t  up  u])on 
two  different  ])lanes  of  abstraction;  physiological 
space  includes,  and  mathematical  space  excludes, 
the  sensory  data  of  juxtaposition.  T^hysiological 
space  admits  concrete  facts, — man's  own  upright 
position,  gravity,  perspective,  etc.  Mathematical 
space  is  purely  formal,  and  to  lay  its  foundation  we 
ha\c  dug  down  to  the  bed-rock  of  our  formal  knowl- 
edge, which  is  "anyness."  Mathematical  space  is 
a  priori,  albeit  the  a  priori  of  niDtion. 

At  present  it  is  sufficient  to  state  that  the  homo- 
geneity of  a  mathematical  space  is  its  anyncss,  and 


68  THE   FOUNDATIONS   OF   MATHEMATICS. 

its  anyness  is  due  to  our  construction  of  it  in  the 
domain  of  pure  form,  involving  universality  and 
excluding  everything  concrete  and  particular. 

The  idea  of  homogeneity  in  our  space-conception 
is  the  tacit  condition  for  the  theorems  of  similarity 
and  proportion,  and  also  of  free  mobility  without 
change,  viz.,  that  figures  can  be  shifted  about  with- 
out suffering  distortion  either  by  shrinkage  or  by 
expanse.  The  principle  of  homogeneity  being  ad- 
mitted, we  can  shove  figures  about  on  any  surface 
the  curvature  of  which  is  either  constant  or  zero. 
This  produces  either  the  non-Euclidean  geometries 
of  spherical,  pseudo-spherical,  and  elliptic  surfaces, 
or  the  plane  geometry  of  Euclid — all  of  them  a 
priori  constructions  made  without  reference  to  real- 
ity. 

Our  a  priori  constructions  serve  an  important 
purpose.  We  use  them  as  systems  of  reference. 
We  construct  a  priori  a  number  system,  making  a 
simple  progression  through  a  series  of  units  which 
we  denominate  from  the  starting-point  o,  as  i,  2, 
3,  4,  5,  6,  etc.  These  numbers  are  purely  ideal  con- 
structions, but  with  their  help  we  can  count  and 
measure  and  weigh  the  several  objects  of  reality 
that  confront  experience;  and  in  all  cases  we  fall 
back  upon  our  ideal  number  system,  saying,  the 
table  has  four  legs ;  it  is  two  and  a  half  feet  high,  it 
weighs  fifty  pounds,  etc.  We  call  these  modes  of 
determination  quantitative. 

The  element  of  quantitative  measurement  is  the 
ideal  construction  of  units,  all  of  which  are  assumed 


THE  PHILOSOPHICAL   BASIS.  69 

to  be  discrete  and  equivalent.  The  equivalence  of 
numbers  as  much  as  the  homogeneity  of  space,  is 
due  to  abstraction.  In  reality  equivalent  units  do 
not  exist  any  more  than  different  parts  of  real  space 
may  be  regarded  as  homogeneous.  Both  construc- 
tions have  been  made  to  create  a  domain  of  anyness, 
for  the  purpose  of  standards  of  reference. 

EVEN  BOUXDARIES  AS  STANDARDS  OF  MEASUREMENT. 

Standards  of  reference  are  useful  only  when 
they  are  unique,  and  thus  we  cannot  use  any  path 
of  our  spread  of  motion  in  all  directions,  but  must 
select  one  that  admits  of  no  equivocation.  The  only 
line  that  possesses  this  quality  is  the  ray,  viz.,  the 
straight  line  or  the  path  of  greatest  intensity. 

The  straight  line  is  one  instance  only  of  a  whole 
class  of  similar  constructions  which  with  one  name 
may  be  called  ''even  boundaries,"  and  by  even  I 
mean  congruent  with  itself.  They  remain  the  same 
in  any  position  and  no  change  originates  however 
they  may  be  turned. 

Clifford,  starting  from  objective  space,  con- 
structs the  plane  by  polishing  three  surfaces.  A,  B, 
and  C,  until  Ihcy  fit  one  another,  which  means  until 
they  are  congruent."'  TTis  proposition  leads  to  the 
same  result  as  ours,  but  the  essential  thing  is  not 
so  much  fas  Clifford  has  it)  that  the  three  planes 
are  congruent,  each  to  the  two  others,  but  that  earli 
plane  is  congruent  with  its  own  inversion.     'I'Iuls, 

'*  Common  Soisr  of  the  lixnrl  Sciniccs,  Appleton  &  Co.,  p.  66. 


/O  THE   FOUNDATIONS  OF   MATHEMATICS. 

under  all  conditions,  each  one  is  congruent  with 
itself.  Each  plane  partitions  the  whole  infinite  space 
into  two  congruent  halves. 

Having  divided  space  so  as  to  make  the  boun- 
dary surface  congruent  with  itself  (viz.,  a  plane), 
we  now^  divide  the  plane  (we  will  call  it  P)  in  the 
same  way, — a  process  best  exemplified  in  the  fold- 
ing of  a  sheet  of  paper  stretched  flat  on  the  table. 
The  crease  represents  a  boundary  congruent  with 
itself.  In  contrast  to  curved  lines,  which  cannot  be 
flopped  or  shoved  or  turned  without  involving  a 
change  in  our  construction,  we  speak  of  a  straight 
line  as  an  even  boundary. 

A  circle  can  be  flopped  upon  itself,  but  it  is  not 
an  even  boundary  congruent  with  itself,  because 
the  inside  contents  and  the  outside  surroundings 
are  different. 

If  we  take  a  plane,  represented  by  a  piece  of 
paper  that  has  been  evenly  divided  by  the  crease 
AB,  and  divide  it  again  crosswise,  say  in  the  point 
O,  by  another  crease  CD,  into  two  equal  parts,  we 
establish  in  the  four  angles  round  O  a  new  kind 
of  even  boundary. 

The  bipartition  results  in  a  division  of  each  half 
plane  into  two  portions  which  again  are  congruent 
the  one  to  the  other ;  and  the  line  in  the  crease  CD, 
constituting,  together  with  the  first  crease,  AB,  two 
angles,  is  (like  the  straight  line  and  the  plane) 
nothing  more  nor  less  than  an  even  boundary  con- 
struction.    The  right  angle  originates  by  the  pro- 


THE  PHILOSOPHICAL  BASIS.  7 1 

cess  of  halving  the  straight  Hue  conceived  as  an 
angle. 

Let  us  now  consider  the  significance  of  e\en 
boundaries. 

A  point  being  a  mere  locus  in  space,  has  no  ex- 
tension whatever:  it  is  congruent  with  itself  on 
account  of  its  want  of  any  discriminating  parts.  If 
it  rotates  in  any  direction,  it  makes  no  difference. 

There  is  no  mystery  about  a  point's  being  con- 
gruent with  itself  in  any  position.  It  results  from 
our  conception  of  a  point  in  agreement  with  the  ab- 
straction we  have  made;  but  when  we  are  con- 
fronted with  lines  or  surfaces  that  are  congruent 
with  themselves  w^e  believe  ourselves  nonplussed; 
yet  the  mystery  of  a  straight  line  is  not  greater  than 
that  of  a  point. 

A  line  which  when  flopped  or  turned  in  its  direc- 
tion remains  congruent  with  itself  is  called  straight, 
and  a  surface  which  when  flopped  or  turned  round 
on  itself  remains  congruent  with  itself  is  called 
])lane  or  flat. 

The  straight  lines  and  the  flat  surfaces  are, 
among  all  jjossible  boundaries,  of  special  im])or- 
tance,  for  a  similar  reason  that  the  abstraction  of 
])ure  form  is  so  useful.  In  tlic  domain  of  pure  form 
we  get  rid  of  all  particularity  and  thus  establish 
a  norm  fit  for  universal  application.  In  geometry 
straight  lines  and  plane  surfaces  are  the  climax 
of  simplicity;  they  are  void  of  any  particularity 
that  needs  further  description,  or  would  complicate 
tlie  situation,  and  this  absence  of  complications  in 


72  THE   FOUNDATIONS   OF   MATHEMATICS. 

their  construction  is  their  greatest  recommendation. 
The  most  important  point,  however,  is  their  qiiaHty 
of  being  unique  by  being  even.  It  renders  them 
specially  available  for  purposes  of  reference. 

We  can  construct  a  priori  different  surfaces  that 
are  homogeneous,  yielding  as  many  different  sys- 
tems of  geometry.  Euclidean  geometry  is  neither 
more  nor  less  true  than  spherical  or  elliptic  geom- 
etry; all  of  them  are  purely  formal  constructions, 
they  are  a  priori,  being  each  one  on  its  own  premises 
irrefutable  by  experience;  but  plane  geometry  is 
more  practical  for  general  purposes. 

The  question  in  geometry  is  not,  as  some  meta- 
geometricians  would  have  it,  "Is  objective  space 
flat  or  curved?"  but,  "Is  it  possible  to  make  con- 
structions that  shall  be  unique  so  as  to  be  service- 
able as  standards  of  reference?"  The  former  ques- 
tion is  due  to  a  misconception  of  the  nature  of  math- 
ematics ;  the  latter  must  be  answered  in  the  affirma- 
t\Yt.  All  even  boundaries  are  unique  and  can  there- 
fore be  used  as  standards  of  reference. 


THE    STRAIGHT    LINE    INDISPENSABLE. 

Straight  lines  do  not  exist  in  reality.  How 
rough  are  the  edges  of  the  straightest  rulers,  and 
how  rugged  are  the  straightest  lines  drawn  with 
instruments  of  precision,  if  measured  by  the  stan- 
dard of  mathematical  straightness !  And  if  we  con- 
sider the  paths  of  motion,  be  they  of  chemical  atoms 
or  terrestrial  or  celestial  bodies,  we  shall  alwavs 


THE  PHILOSOPHICAL   BASIS.  'J}^ 

find  them  to  be  curves  of  hig-h  complexity.  Never- 
theless the  idea  of  the  straight  line  is  justified  by 
experience  in  so  far  as  it  helps  us  to  analyze  the 
complex  curves  into  their  elementary  factors,  no 
one  of  which  is  truly  straight;  but  each  one  of 
which,  when  we  go  to  the  end  of  our  analysis,  can 
be  represented  as  a  straight  line.  Judging  from 
the  experience  we  have  of  moving  bodies,  we  cannot 
doubt  that  if  the  sun's  attraction  of  the  earth  (as 
well  as  that  of  all  other  celestial  bodies)  could  be 
annihilated,  the  earth  would  fly  off  into  space  in 
a  straight  line.  Thus  the  mud  on  carriage  wheels, 
when  spurting  off,  and  the  pebbles  that  are  thrown 
with  a  sling,  are  flying  in  a  tangential  direction 
which  would  be  absolutely  straight  were  it  not  for 
the  interference  of  the  gravity  of  the  earth,  which 
is  constantly  asserting  itself  and  modifies  the 
straightest  line  into  a  curve. 

Otir  idea  of  a  straight  line  is  suggested  to  us  by 
experience  when  we  attem])t  to  resolve  comp(umd 
forces  into  their  constituents,  but  it  is  not  traceable 
in  experience.  It  is  a  product  of  our  method  of 
measurement.  It  is  a  creation  of  our  own  doings, 
yet  it  is  justified  by  the  success  which  attends  '\\< 
employment. 

The  great  question  in  geometry  is  not.  whether 
straight  lines  are  real  but  whether  their  construc- 
tion is  not  an  indispensable  ref|uisite  for  any  pos- 
sible system  of  space  mcasurctncnl.  nnd  fnrtlier, 
what  is  the  nature  of  straight  lines  and  planes  and 
right  angles;  how  does  their  conce])tion  originate 


74  THE   FOUNDATIONS  OF   MATHEMATICS. 

and  why  are  they  of  paramount  importance  in  ge- 
ometry. 

We  can  of  course  posit  that  space  should  be 
filled  up  with  a  medium  such  as  would  deflect  every 
ray  of  light  so  that  straight  rays  would  be  impos- 
sible. For  all  we  know  ether  may  in  an  extremely 
slight  degree  operate  in  that  way.  But  there  would 
be  nothing  in  that  that  could  dispose  of  the  think- 
ability  of  a  line  absolutely  straight  in  the  Euclidean 
sense  with  all  that  the  same  involves,  so  that  Eu- 
clidean geometry  would  not  thereby  be  invalidated. 

Now  the  fact  that  the  straight  line  (as  a  purely 
mental  construction)  is  possible  cannot  be  denied: 
we  use  it  and  that  should  be  sufficient  for  all  prac- 
tical purposes.  That  we  can  construct  curves  also 
does  not  invalidate  the  existence  of  straight  lines. 

So  again  while  a  geometry  based  upon  the  idea 
of  homaloidal  space  will  remain  what  it  has  ever 
been,  the  other  geometries  are  not  made  thereby  il- 
legitimate. Euclid  disposes  as  little  of  Lobachevsky 
and  Bolyai  as  they  do  of  Euclid. 

As  to  the  nature  of  the  straight  line  and  all  the 
other  notions  connected  therewith,  we  shall  always 
be  able  to  determine  them  as  concepts  of  boundary, 
either  reaching  the  utmost  limit  of  a  certain  func- 
tion, be  it  of  the  highest  (such  as  » )  or  lowest  meas- 
ure (such  as  o) ;  or  dividing  a  whole  into  two  con- 
gruent parts. 

The  utility  of  such  boundary  concepts  becomes 
apparent  when  we  are  in  need  of  standards  for 
measurement.    An  even  boundary  being  the  utmost 


THE  PHILOSOPHICAL   BASIS. 


/ :) 


limit  is  unique.  There  are  innumerable  curves,  but 
there  is  only  one  kind  of  straight  line.  Accordingly, 
if  we  need  a  standard  for  measuring  curves,  we 
must  naturally  fall  back  upon  the  straight  line  and 
determine  its  curvature  by  its  deviation  from  the 
straight  line  which  represents  a  zero  of  curvature. 

The  straight  line  is  the  simplest  of  all  boundary 
concepts.     Hence  its  indispensableness. 

If  we  measure  a  curvature  we  resolve  the  curve 
into  infinitesimal  pieces  of  straight  lines,  and  then 
determine  their  change  of  direction.  Thus  we  use 
the  straight  line  as  a  reference  in  our  measurement 
of  curves.  The  simplest  curve  is  the  circle,  and  its 
curvature  is  expressed  by  the  reciprocal  of  the  ra- 
dius; but  the  radius  is  a  straight  line.  It  seems 
that  we  cannot  escape  straightness  anywhere  in 
geometry;  for  it  is  the  simplest  instrument  for  meas- 
uring distance.  We  may  replace  metric  geometry 
by  projective  geometry,  but  what  could  projective 
geometricians  do  if  they  had  not  straight  lines  for 
their  projections'  Without  them  they  would  be 
in  a  strait  indeed! 

But  suppose  we  renounced  with  Lobatchevsky 
the  conventional  method  of  even  boundary  concep- 
tions, especially  straightness  of  line,  and  were  satis- 
fied with  straightest  lines,  what  would  be  the  result? 
Tie  does  not  at  the  same  time,  surrender  either  the 
principle  of  consistency  or  the  assumption  of  the 
homogeneity  of  space,  and  thus  he  builds  up  a  ge- 
ometry independent  of  the  theorem  of  ])arallel  lines, 
which  would  be  applicable  to  two  systems,  the  Eu- 


76  THE   FOUNDATIONS  OF   MATHEMATICS. 

clidean  of  straight  lines  and  the  non-Eudidean  of 
curved  space.  But  the  latter  needs  the  straight  line 
as  much  as  the  former  and  finds  its  natural  limit 
in  a  sphere  whose  radius  is  infinite  and  whose  curva- 
ture is  zero.  He  can  measure  no  spheric  curvature 
without  the  radius,  and  after  all  he  reaches  the 
straight  line  in  tlie  limit  of  curvature.  Yet  it  is 
noteworthy  that  in  the  Euclidean  system  the  straight 
line  is  definite  and  tt  irrational,  while  in  the  non- 
Euclidean,  77  is  a  definite  number  according  to  the 
measure  of  curvature  and  the  straight  line  becomes 
irrational. 


THE  SUPERREAL. 

We  said  in  a  former  chapter  (p.  48),  "man  did 
not  invent  reason,  he  discovered  it,"  which  means 
that  the  nature  of  reason  is  definite,  unalterable, 
and  therefore  valid.  The  same  is  true  of  all  any- 
ness  of  all  formal  thought,  of  pure  logic,  of  mathe- 
matics, and  generally  of  anything  that  with  truth 
can  be  stated  a  priori.  Though  the  norms  of  any- 
ness  are  woven  of  pure  nothingness,  the  flimsiest 
material  imaginable,  they  are  the  factors  which  de- 
termine the  course  of  events  in  the  entire  sweep 
of  actual  existence  and  in  this  sense  they  are  real. 
They  are  not  real  in  the  sense  of  materiality;  they 
are  real  only  in  being  efficient  and  in  distinction 
to  the  reality  of  corporeal  things  we  may  call  them 
superreal. 

On  the  one  hand  it  is  true  that  mathematics  is 


THE  PHILOSOPHICAL   BASIS.  ']'] 

a  mental  construction ;  it  is  purely  ideal,  which  means 
it  is  \vo\'en  of  thought.  On  the  other  hand  we  must 
grant  that  the  nature  of  this  construction  is  fore- 
determined  in  its  minutest  detail  and  in  this  sense 
all  its  theorems  must  be  discovered.  \\'e  grant 
that  there  are  no  sines,  and  cosines,  no  tangents 
and  cotangents,  no  logarithms,  no  number  tt,  nor 
even  lines,  in  nature,  but  there  are  relations  in 
nature  which  correspond  to  these  notions  and  sug- 
gest the  invention  of  symbols  for  the  sake  of  de- 
termining them  with  exactness.  These  relations 
possess  a  normative  value.  Stones  are  real  in  the 
sense  of  offering  resistance  in  a  special  ])lace.  but 
these  norms  are  superreal  because  they  are  efiicient 
factors  everywhere. 

The  reality  of  mathematics  is  well  set  forth  in 
these  words  of  Prof.  Cassius  Jackson  Keyser,  of 
Columbia  University : 

■'Phrase  it  as  you  will,  there  is  a  world  that  is  peopled 
with  ideas,  ensembles,  i)ropositions,  relations,  and  implica- 
tions, in  endless  variety  and  multiplicity,  in  structure  ran- 
i^ing-  from  the  very  simple  to  the  endlessly  intricate  and  com- 
plicate. That  world  is  not  the  product  but  the  object,  not 
the  creature  but  the  quarry  of  thought,  the  entities  compos- 
ing it — propositions,  for  example. — being  no  more  identical 
with  thinking  them  than  wine  is  identical  with  the  drinking 
of  it.  Mind  or  no  mind,  lliat  world  exists  as  an  extra- 
personal  affair. — pragmatism  to  tlu'  contrarv  notwithstand- 
ing." 

\\  bile  the  relational  possesses  objective  signifi- 
cance, the  mctluifl  of  (k'scri])ing  il  is  subjective  and 
<')f  course  the  symbols  .-irc  .'irbilrarv. 


78  THE   FOUNDATIONS  OF   MATHEMATICS. 

In  this  connection  we  wish  to  call  attention  to  a 
most  important  point,  which  is  the  necessity  of  cre- 
ating fixed  miits  for  counting.  As  there  are  no 
logarithms  in  nature,  so  there  are  no  numbers; 
there  are  only  objects  or  things  sufficiently  equal 
which  for  a  certain  purpose  may  be  considered 
equivalent,  so  that  we  can  ignore  these  dififerences, 
and  assuming  them  to  be  the  same,  count  them. 

DISCRETE  UNITS   AND  THE   CONTINUUM. 

Nature  is  a  continuum ;  there  are  no  boundaries 
among  things,  and  all  events  that  happen  proceed 
in  an  uninterrupted  flow  of  continuous  transforma- 
tions. For  the  sake  of  creating  order  in  this  flux 
which  would  seem  to  be  a  chaos  to  us,  we  must 
distinguish  and  mark  off  individual  objects  with 
definite  boundaries.  This  method  may  be  seen  in  all 
branches  of  knowledge,  and  is  most  in  evidence  in 
arithmetic.  When  counting  we  start  in  the  domain 
of  nothingness  and  build  up  the  entire  structure  of 
arithmetic  with  the  products  of  our  own  making. 

We  ought  to  know  that  w^hatever  we  do,  we  must 
first  of  all  take  a  definite  stand  for  ourselves.  When 
we  start  doing  anything,  we  must  have  a  starting- 
point,  and  even  though  the  world  may  be  a  constant 
flux  we  must  for  the  sake  of  definiteness  regard  our 
starting-point  as  fixed.  It  need  not  be  fixed  in  real- 
ity, but  if  it  is  to  serve  as  a  point  of  reference  we 
must  regard  it  as  fixed  and  look  upon  all  the  rest 
as  movable;    otherwise  the  world  would  be  an  in- 


THE  PHILOSOPHICAL   BASIS.  79 

determinable  tangle.  Here  we  have  the  first  rule 
of  mental  activity.  There  may  be  no  rest  in  the 
world  yet  we  must  create  the  fiction  of  a  rest  as  a 
809  /xot  TTov  (TTco  aud  whcuever  we  take  any  step 
we  must  repeat  this  fictitious  process  of  laying 
down  definite  points. 

All  the  things  which  are  observed  around  us  are 
compounds  of  qualities  which  are  only  temporarily 
combined.  To  call  them  things  as  if  they  were 
separate  beings  existing  without  reference  to  the 
rest  is  a  fiction,  but  it  is  part  of  our  method  of 
classification,  and  without  this  fictitious  comprehen- 
sion of  certain  groups  of  qualities  under  definite 
names  and  treating  them  as  units,  we  could  make 
no  headway  in  this  world  of  constant  flux,  and  all 
events  of  life  would  swim  before  our  mental  eye. 

Our  method  in  arithmetic  is  similar.  We  count 
as  if  units  existed,  yet  the  idea  of  a  unit  is  a  fiction. 
We  count  our  fingers  or  the  beads  of  an  abacus  or 
any  other  set  of  things  as  if  they  were  equal.  We 
count  the  feet  which  we  measure  off  in  a  certain 
line  as  if  each  one  were  equivalent  to  all  the  rest. 
For  all  we  know  they  may  be  different,  but  for  our 
])urpose  of  measuring  they  possess  the  same  signifi- 
cance. This  is  neither  an  hypothesis  nor  an  assump- 
tion uov  a  fiction,  but  a  postulate  needed  for  a 
definite  purpfjsc.  l-'or  our  purpose  and  according 
to  tile  method  eniploNcd  they  are  the  same.  We 
])Ostulate  their  sameness.  We  have  made  tliem  the 
same,  we  treat  them  as  equal.    Their  sameness  de- 


8o  THE   FOUNDATIONS  OF   MATHEMATICS. 

pends  upon  the  conditions  from  which  we  start  and 
on  the  purpose  which  we  have  in  view. 

There  are  theorems  which  are  true  in  arithmetic 
but  which  do  not  hold  true  in  practical  life.  I  will 
only  mention  the  theorem  2-}-34-4  =  4-I-3+2  = 
4+2+3  etc.  In  real  life  the  order  in  which  things 
are  pieced  together  is  sometimes  very  essential,  but 
in  pure  arithmetic,  when  we  have  started  in  the 
domain  of  nothingness  and  build  with  the  products 
of  our  own  counting  which  are  ciphers  absolutely 
equivalent  to  each  other,  the  rule  holds  good  and 
it  will  be  serviceable  for  us  to  know  it  and  to  utilize 
its  significance. 

The  positing  of  units  which  appears  to  be  an 
indispensable  step  in  the  construction  of  arithmetic 
is  also  of  great  importance  in  actual  psychology 
and  becomes  most  apparent  in  the  mechanism  of 
vision. 

Consider  the  fact  that  the  kinematoscope  has 
become  possible  only  through  an  artificial  separation 
of  the  successive  pictures  which  are  again  fused 
together  into  a  new  continuum.  The  film  which 
passes  before  the  lens  consists  of  a  series  of  little 
pictures,  and  each  one  is  singly  presented,  halting 
a  moment  and  being  separated  from  the  next  by  a 
rotating  fan  which  covers  it  at  the  moment  when  it 
is  exchanged  for  the  succeeding  picture.  If  the 
moving  figures  on  the  screen  did  not  consist  of  a 
definite  number  of  pictures  fused  into  one  by  our 
eye  which  is  incapable  of  distinguishing  their  quick 
succession  the  whole  sisfht  would  be  blurred  and  we 


THE  PHILOSOPHICAL  BASIS.  8l 

could  see  nothing  but  an  indiscriminate  and  un- 
analyzable  perpetual  flux. 

This  method  of  our  mind  which  produces  units 
in  a  continuum  may  possess  a  still  deeper  signifi- 
cance, for  it  may  mark  the  very  beginning  of  the 
real  world.  For  all  we  know  the  formation  of  the 
chemical  atoms  in  the  evolution  of  stellar  nebulas 
may  be  nothing  but  an  analogy  to  this  process.  The 
manifestation  of  life  too  begins  with  the  creation 
of  individuals — of  definite  living  creatures  which 
develop  differently  under  different  conditions  and 
again  the  soul  becomes  possible  by  the  definiteness 
of  single  sense-impressions  which  can  be  distin- 
guished as  units  from  others  of  a  different  type. 

Thus  the  contrast  between  the  continuum  and 
the  atomic  formation  appears  to  be  fundamental 
and  gives  rise  to  many  problems  which  have  be- 
come especially  troublesome  in  mathematics.  But 
if  we  bear  in  mind  that  the  method,  so  to  speak,  of 
atomic  division  is  indispensable  to  change  a  world 
of  continuous  flux  into  a  system  that  can  be  com- 
puted and  determined  with  at  least  approximate 
accuracy,  we  will  be  apt  to  appreciate  that  the 
atomic  fiction  in  arilhiiu'tic  is  an  indis])cnsablc  jiart 
of  the  method  bv  w  hich  liie  whole  science  is  created. 


MATHEMATICS    AND    METAGEOMETRY. 

DIFFERENT  GEOMETRICAL  SYSTEMS. 

STRAIGHTNESS,  flatness,  and  rectangularity 
are  qualities  which  cannot  (Hke  numbers)  be 
determined  in  purely  quantitative  terms;  but  they 
are  determined  nevertheless  by  the  conditions  under 
which  our  constructions  must  be  made.  A  right 
angle  is  not  an  arbitrary  amount  of  ninety  degrees, 
but  a  quarter  of  a  circle,  and  even  the  nature  of 
angles  and  degrees  is  not  derivable  either  from 
arithmetic  or  from  pure  reason.  They  are  not  purely 
quantitative  magnitudes.  They  contain  a  qualita- 
tive element  which  cannot  be  expressed  in  numbers 
alone.  A  plane  is  not  zero,  but  a  zero  of  curvature 
in  a  boundary  between  two  solids ;  and  its  qualita- 
tive element  is  determined,  as  Kant  would  express 
it,  by  Anschauung,  or  as  we  prefer  to  say,  by  pure 
motility,  i.  e.,  it  belongs  to  the  domain  of  the  a 
priori  of  doing.  For  Kant's  term  Auschaming  has 
the  disadvantage  of  suggesting  the  passive  sense 
denoted  by  the  word  "contemplation,"  while  it  is 
important  to  bear  in  mind  that  the  thinking  subject 
by  its  own  activities  creates  the  conditions  that  de- 
termine the  qualities  alwve  mentioned. 


MATHEMATICS  AND   METAGEOMETRV 


«3 


Our  method  of  creating  by  construction  the 
straight  Hne,  the  plane,  and  the  right  angle,  does 
not  exclude  the  possibility  of  other  methods  of 
space-measurement,  the  standards  of  which  would 
not  be  even  boundaries,  such  as  straight  lines,  but 
lines  possessed  of  either  a  positive  curvature  like 
the  sphere  or  a  negative  curvature  rendering  their 
surface  pseudo-spherical. 

Spheres  are  well  known  and  do  not  stand  in 
need  of  description.  Their  curvature  which  is  posi- 
tive is  determined  by  the  reciprocal  of  their  radius. 

A 


B 

Fig.  I. 


Pseudo-Spheres  are  surfaces  of  negative  curva- 
ture, and  pseudo-spherical  surfaces  are  saddle- 
shaped.  Only  limited  pieces  can  1)e  connectedly 
represented,  and  we  reproduce  from  llelmhoUz,' 
two  instances.  If  arc  ah  in  figure  i  revolves  round 
an  axis  A]>.  it  will  describe  a  concave-convex  sur- 
face like  that  of  the  inside  of  a  wedding-ring;  and 
in  the  same  way.  if  cither  of  the  curves  of  figure  2 
revolve  round  their  axis  of  symmetry,  it  will  de- 
scribe one  lialf  of  a  pseudospherical  surface  rcseni- 

'  Loc.  cit.,  p.  42. 


84  THE   FOUNDATIONS  OF   MATHEMATICS. 

bling  the  shape  of  a  morning-glory  whose  tapering 
stem  is  infinitely  prolonged.  Helmholtz  compares 
the  former  to  an  anchor-ring,  the  latter  to  a  cham- 
pagne glass  of  the  old  style. 

The  sum  of  the  angles  of  triangles  on  spheres 
always  exceeds  i8o°,  and  the  larger  the  sphere  the 
more  will  their  triangles  resemble  the  triangle  in 
the  plane.  On  the  other  hand,  the  sum  of  the  angles 
of  triangles  on  the  pseudosphere  will  always  be 
somewhat  less  than  i8o°.  If  we  define  the  right 
angle  as  the  fourth  part  of  a  whole  circuit,  it  will 
be  seen  that  analogously  the  right  angle  in  the 
plane  differs  from  the  right  angles  on  the  sphere  as 
well  as  the  pseudosphere. 

We  may  add  that  while  in  spherical  space  sev- 
eral shortest  lines  are  possible,  in  pseudospherical 
space  we  can  draw  one  shortest  line  only.  Both  sur- 
faces, however,  are  homogeneous  (i.  e.,  figures  can 
be  moved  in  it  without  sufifering  a  change  in  dimen- 
sions), but  the  parallel  lines  w^iich  do  not  meet  are 
impossible  in  either. 

Wt  may  further  construct  surfaces  in  w^hich 
changes  of  place  involve  either  expansion  or  con- 
traction, but  it  is  obvious  that  they  would  be  less 
serviceable  as  systems  of  space-measurement  the 
more  irregular  they  grow. 

TRIDIMENSIONALITY. 

Space  is  usually  regarded  as  tridimensional,  but 
there  are  some  people  who,  following  Kant,  express 


MATHEMATICS  AND   METAGEOMETRY.  85 

themselves  with  reserve,  saying  that  the  mind  of 
man  ma}-  be  buih  u|)  in  such  a  way  as  to  conceive 
of  objects  in  terms  of  three  cHmensions.  Others 
think  that  the  actual  and  real  thing  that  is  called 
space  may  be  quite  different  from  our  tridimensional 
conception  of  it  and  ma}-  in  point  of  fact  be  four, 
or  five,  or  //-dimensional. 

Let  us  ask  first  what  "dimension"  means. 

Does  dimension  mean  direction?  Obviously  not, 
for  we  have  seen  that  the  possibilities  of  direction  in 
space  are  infinite. 

Dimension  is  only  a  popular  term  for  co-ordi- 
nate. In  space  there  are  no  dimensions  laid  down, 
but  in  a  space  of  infinite  directions  three  co-ordinates 
are  needed  to  determine  from  a  gi\en  point  of  ref- 
erence the  position  of  any  other  point. 

In  a  former  section  on  "Even  Boundaries  as 
Standards  of  Measurement,''  we  have  halved  space 
and  produced  a  plane  Pi  as  an  e\en  boundary  be- 
tween the  two  halves ;  we  have  halved  the  plane  Pi 
by  turning  the  plane  so  upon  itself,  that  like  a  crease 
in  a  folded  sheet  of  paper  the  straight  line  AB 
was  produced  on  the  plane.  We  then  halved  the 
straight  line,  the  even  boundary  between  the  two 
half-planes,  by  again  turning  the  plane  upon  itself 
so  that  the  line  AB  covered  its  own  prolongation. 
It  is  as  if  our  folded  sheet  of  paper  were  folded  a 
second  time  u])r)n  itself  so  that  the  crease  would 
be  folded  upon  itself  and  one  part  of  the  same  fall 
exactly  upon  and  cover  the  other  part.  On  opening 
the  sheet  we  have  a  second  crease  crossing  the  first 


86  THE   FOUNDATIONS  OF   MATHEMATICS. 

one  making  the  perpendicular  CD,  in  the  point  O, 
thus  producing  right  angles  on  the  straight  line 
AB,  represented  in  the  cross-creases  of  the  twice 
folded  sheet  of  paper.  Here  the  method  of  producing 
even  boundaries  by  halving  comes  to  a  natural  end. 
So  far  our  products  are  the  plane,  the  straight  line, 
the  point  and  as  an  incidental  but  valuable  by- 
product, the  right  angle. 

We  may  now  venture  on  a  synthesis  of  our  ma- 
terials. We  lay  two  planes,  Po  and  P?.,  through  the 
two  creases  at  right  angles  on  the  original  plane 
Pi,  represented  by  the  sheet  of  paper,  and  it  becomes 
apparent  that  the  two  new  planes  Po  and  P3  wnll 
intersect  at  O,  producing  a  line  EF  common  to  both 
planes  P2  and  P3,  and  they  will  bear  the  same  rela- 
tion to  each  as  each  one  does  to  the  original  plane 
Pi,  that  is  to  say:  the  whole  system  is  congruent 
with  itself.  If  we  make  the  planes  change  places,  Pi 
may  as  well  take  the  place  of  Po  and  P2  of  P3  and 
Po  of  P2  or  Pi  of  P.3,  etc.,  or  vice  versa,  and  all  the 
internal  relations  would  remain  absolutely  the  same. 
Accordingly  we  have  here  in  this  system  of  the 
three  planes  at  right  angles  (the  result  of  repeated 
halving),  a  composition  of  even  boundaries  which, 
as  the  simplest  and  least  complicated  construction 
of  its  kind,  recomends  itself  for  a  standard  of  meas- 
urement of  the  w'hole  spread  of  motility. 

The  most  significant  feature  of  our  construction 
consists  in  this,  that  we  thereby  produce  a  con- 
venient system  of  reference  for  determining  every 


MATHEMATICS  AND   METAGEOMETRY.  87 

possible  point  in  co-ordinates  of  straight  lines  stand- 
ing at  right  angles  to  the  three  planes. 

If  we  start  from  the  ready  conception  of  objec- 
tive space  (the  juxtaposition  of  things)  we  can 
refer  the  several  distances  to  analogous  loci  in  our 
system  of  the  three  planes,  mutually  perpendicular, 
each  to  the  others.  We  cut  space  in  two  equal  halves 
by  the  horizontal  plane  Pj.  We  repeat  the  cutting  so 
as  to  let  the  two  halves  of  the  first  cut  in  their  angu- 
lar relation  to  the  new  cut  (in  Po)  be  congruent 
with  each  other,  a  procedure  which  is  possible  only 
if  we  make  use  of  the  even  boundary  concept  with 
which  we  have  become  acquainted.  Accordingly, 
the  second  cut  should  stand  at  right  angles  on  the 
first  cut.  The  two  planes  Pi  and  P2  have  one  line 
in  common,  EF,  and  any  plane  placed  at  right  an- 
gles to  EF  (in  the  point  O)  will  again  satisfy  the 
demand  of  dividing  space,  including  the  two  planes 
Pi  and  P2,  into  two  congruent  halves.  The  two 
new  lines,  produced  by  the  cut  of  the  third  plane 
P..  through  the  two  former  planes  Pi  and  Po.  stand 
both  at  right  angles  to  EF.  Should  we  continue 
our  method  of  cutting  space  at  right  angles  in  O 
on  either  of  these  lines,  we  would  produce  a  ])lanc 
coincident  with  Pi,  which  is  to  say,  that  ihc  ])ossi- 
bilities  of  the  system  are  exhausted. 

This  implies  that  in  any  system  of  pure  space 
flircc  co-nriihutlcs  arc  sufliciciit  iov  the  determina- 
tion of  an\-  place  from  a  given  reference  i)()int. 


88  THE  FOUNDATIONS  OF  MATHEMATICS. 


THREE  A  CONCEPT  OF  BOUNDARY. 

The  number  three  is  a  concept  of  boundary  as 
much  as  the  straight  line.  Under  specially  compli- 
cated conditions  we  might  need  more  than  three 
co-ordinates  to  calculate  the  place  of  a  point,  but  in 
empty  space  the  number  three,  the  lowest  number 
that  is  really  and  truly  a  number,  is  sufficient.  If 
space  is  to  be  empty  space  from  which  the  notion 
of  all  concrete  things  is  excluded,  a  kind  of  model 
constructed  for  the  purpose  of  determining  juxta- 
position, three  co-ordinates  are  sufficient,  because 
our  system  of  reference  consists  of  three  planes, 
and  we  have  seen  above  that  there  is  no  possibility 
of  introducing  a  fourth  plane  without  destroying 
its  character  of  being  congruent  with  itself,  which 
imparts  to  it  the  simplicity  and  uniqueness  that 
render  it  available  for  a  standard  of  measurement. 

Three  is  a  peculiar  number  which  is  of  great 
significance.  It  is  the  first  real  number,  being  the 
simplest  multiplex.  One  and  two  and  also  zero 
are  of  course  numbers  if  we  consider  them  as  mem- 
bers of  the  number-system  in  its  entirety,  but  singly 
regarded  they  are  not  yet  numbers  in  the  full  sense 
of  the  word.  One  is  the  unit,  two  is  a  couple  or  a 
pair,  but  three  is  the  smallest  amount  of  a  genuine 
plurality.  Savages  who  can  distinguish  only  be- 
tween one  and  two  have  not  yet  evolved  the  notion 
of  number;  and  the  transition  to  the  next  higher 
stage  involving  the  knowledge  of  "three"  passes 
through  a  mental  condition  in  which  there  exists 


MATHEMATICS  AND   METAGEOMETRY.  8g 

only  the  notion  one,  two,  and  plurality  of  any  kind. 
When  the  idea  of  three  is  once  definitely  recognized, 
the  naming-  of  all  other  numbers  can  follow  in  rapid 
succession. 

In  this  connection  we  may  incidentally  call  at- 
tention to  the  significance  of  the  grammatical  dual 
number  as  seen  in  the  Semitic  and  Greek  languages. 
It  is  a  surviving  relic  and  token  of  a  period  during 
which  the  unit,  the  pair,  and  the  uncounted  plural- 
ity constituted  the  entire  gamut  of  human  arith- 
metic. The  dual  form  of  grammatical  number  by 
the  development  of  the  nimiber-system  became  re- 
dundant and  cumbersome,  being  retained  only  for 
a  while  to  express  the  idea  of  a  couple,  a  pair  that 
naturally  belong  together. 

Certainly,  the  origin  of  the  notion  three  has  its 
germ  in  the  nature  of  abstract  anyness.  Nor  is 
it  an  accident  that  in  order  to  construct  the  simplest 
figure  which  is  a  real  figure,  at  least  three  lines 
are  needed.  The  importance  of  the  triangle,  which 
becomes  most  prominent  in  trigonometry,  is  due  to 
its  being  the  simplest  i)ossil)lc  figure  which  accord- 
ingly possesses  the  intrinsic  worth  of  economy. 

The  number  three  plays  also  a  significant  part 
in  logic,  and  in  the  branches  of  the  a])plied  sciences, 
and  thus  we  need  not  be  astonished  at  finding  the 
very  idea,  three,  Iield  in  religious  reverence,  for  the 
doctrine  of  tlie  Trinil\-  has  its  l)asis  in  the  constitu- 
tion of  the  universe  and  can  be  fnll\-  justified  ])\-  the 
laws  of  pure  form. 


go  THE  FOUNDATIONS  OF   MATHEMATICS. 

SPACE  OF  FOUR  DIMENSIONS. 

The  several  conceptions  of  space  of  more  than 
three  dimensions  are  of  a  purely  abstract  nature, 
yet  they  are  by  no  means  vague,  but  definitely  de- 
termined by  the  conditions  of  their  construction. 
Therefore  we  can  determine  their  properties  even 
in  their  details  with  perfect  exactness  and  formu- 
late in  abstract  thought  the  laws  of  four-,  five-,  six-, 
and  7?-dimensional  space.  The  difficulty  with  which 
we  are  beset  in  constructing  7z-dimensional  spaces 
consists  in  our  inability  to  make  them  representable 
to  our  senses.  Here  we  are  confronted  with  what 
may  be  called  the  limitations  of  our  mental  constitu- 
tion. These  limitations,  if  such  they  be,  are  con- 
ditioned by  the  nature  of  our  mode  of  motion,  which, 
if  reduced  to  a  mathematical  system,  needs  three 
co-ordinates,  and  this  means  that  our  space-con- 
ception is  tridimensional. 

We  ourselves  are  tridimensional ;  we  can  meas- 
ure the  space  in  which  we  move  with  three  co-ordi- 
nates, yet  we  can  definitely  say  that  if  space  were 
four-dimensional,  a  body  constructed  of  two  fac- 
tors, so  as  to  have  a  four-dimensional  solidity,  would 
be  expressed  in  the  formula: 

(a+by=a'+4a^b-{-6a^b--^4ab''+b\ 

We  can  calculate,  compute,  excogitate,  and  de- 
scribe all  the  characteristics  of  four-dimensional 
space,  so  long  as  we  remain  in  the  realm  of  abstract 
thought  and  do  not  venture  to  make  use  of  our 
motility  and  execute  our  plan  in  an  actualized  con- 


MATHEMATICS  AND   METAGEOMETRY. 


91 


struction  of  motion.  From  the  standpoint  of  pure 
logic,  there  is  nothing  irrational  about  the  assump- 
tion; but  as  soon  as  we  make  an  a  priori  construc- 
tion of  the  scope  of  our  motility,  we  find  out  the 
incompatibility  of  the  whole  scheme. 

In  order  to  make  the  idea  of  a  space  of  more  than 
three  dimensions  plausible  or  intelligible,  we  resort 
to  the  relation  between  two-dimensional  beings  and 
tridimensional  space.  The  nature  of  tridimensional 
space  may  be  indicated  yet  not  fully  represented  in 


^ 

^ 

G 

B 

H 

^ 

two-dimensional  space.  If  we  construct  a  square 
upon  the  line  AB  one  inch  long,  it  will  ])e  bounded 
by  four  hues  each  an  inch  in  length.  In  order  to 
construct  upon  the  square  ABCD  a  cube  of  the 
same  measure,  we  must  raise  the  sf|uare  by  one  inch 
into  the  third  dimension  in  a  direction  at  right 
angles  to  its  surface,  tlic  result  being  a  figure 
l)r)undcd  ])v  six  surfaces,  each  of  which  is  a  one-inch 
square.  If  two-dimensional  beings  who  could  not 
rise  into  the  third  dimension  wished  to  gain  an  idea 
of  space  of  a  higher  dimensionality  and  picture  in 


92  THE  FOUNDATIONS  OF   MATHEMATICS. 

their  own  two-dimensional  mathematics  the  results 
of  three  dimensions,  they  might  push  out  the  square 
in  any  direction  within  their  own  plane  to  a  distance 
of  one  inch,  and  then  connect  all  the  corners  of  the 
image  of  the  square  in  its  new  position  with  the 
corresponding  points  of  the  old  square.  The  result 
woiild  be  what  is  to  us  tridimensional  beings  the 
picture  of  a  cube. 

\Mien  we  count  the  plane  quadrilateral  figures 
produced  by  this  combination  we  find  that  there  are 
six,  corresponding  to  the  boundaries  of  a  cube.  We 
must  bear  in  mind  that  only  the  original  and  the  new 
square  will  be  real  squares,  the  four  intermediary 
figures  which  have  originated  incidentally  through 
our  construction  of  moving  the  square  to  a  distance, 
exhibit  a  slant  and  to  our  two-dimensional  beings 
they  appear  as  distortions  of  a  rectangular  relation, 
which  faultiness  has  been  caused  by  the  insuffi- 
ciency of  their  methods  of  representation.  More- 
over all  squares  count  in  full  and  where  their  sur- 
faces overlap  they  count  double. 

Two-dimensional  beings  having  made  such  a 
construction  must  however  bear  in  mind  that  the 
field  covered  by  the  sides,  GEFH  and  BFDH  does 
not  take  up  any  room  in  their  own  plane,  for  it  is 
only  a  picture  of  the  extension  which  reaches  out 
either  above  or  below  their  own  plane;  and  if  they 
venture  out  of  this  field  covered  by  their  construc- 
tion, they  have  to  remember  that  it  is  as  empty  and 
unoccupied  as  the  space  beyond  the  boundaries  AC 
and  AB. 


MATHEMATICS  AND   METAGEOMETRY.  93 

Now  if  we  tridimensional  beings  wish  to  do  the 
same,  how  shall  we  proceed? 

We  must  move  a  tridimensional  body  in  a  rect- 
angular direction  into  a  new  (i.  e.,  the  fourth) 
dimension,  and  being  unable  to  accomplish  this  we 
may  represent  the  operation  by  mirrors.  Having 
three  dimensions  we  need  three  mirrors  standing 
at  right  angles.  We  know  by  a  priori  considera- 
tions according  to  the  principle  of  our  construction 
that  the  boundaries  of  a  four-dimensional  body  must 
be  solids,  i.  e.. tridimensional  bodies,  and  while  the 
sides  of  a  cube  (algebraically  represented  by  a^) 
must  be  six  surfaces  (i.  e.,  two-dimensional  figures, 
one  at  each  end  of  the  dimensional  line)  the  boun- 
daries of  an  analogous  four-dimensional  body  built 
up  like  the  cube  and  the  square  on  a  rectangular 
l)lan,  must  be  eight  solids,  i.  e.,  cubes.  If  we  l)uild 
up  three  mirrors  at  right  angles  and  place  any 
object  in  the  intersecting  corner  we  shall  see  the 
object  not  once,  but  eight  times.  The  body  is  re- 
flected below,  and  the  object  thus  doubled  is  mir- 
rored not  only  on  both  upright  sides  but  in  addition 
in  the  corner  beyond,  appearing  in  citlicr  of  the  up- 
right mirrors  coincidingly  in  the  same  place.  Thus 
tlie  total  multiplication  of  our  tridimensional  boun- 
daries of  a  four-dimensional  complex  is  rendered 
eightfold. 

We  must  now  bear  in  mind  thai  iln's  representa- 
tion of  a  fourth  dimension  suffers  from  all  the  faults 
of  the  analogous  figure  of  a  cube  in  two-dimensional 
space.     'i1ie  several  figures  are  not  eight  indepen- 


94  THE   FOUNDATIONS   OF   MATHEMATICS. 

dent  bodies  but  they  are  mere  boundaries  and  the 
four  dimensional  space  is  conditioned  by  their  inter- 
relation. It  is  that  unrepresentable  something  which 
they  enclose,  or  in  other  words,  of  which  they  are 
assumed  to  be  boundaries.  If  we  were  four-dimen- 
sional beings  we  could  naturally  and  easily  enter 
into  the  mirrored  space  and  transfer  tridimensional 
bodies  or  parts  of  them  into  those  other  objects  re- 
flected here  in  the  mirrors  representing  the  bound- 
aries of  the  four-dimensional  object.  While  thus 
on  the  one  hand  the  mirrored  pictures  would  be  as 
real  as  the  original  object,  they  would  not  take  up 
the  space  of  our  three  dimensions,  and  in  this  re- 
spect our  method  of  representing  the  fourth  dimen- 
sion by  mirrors  would  be  quite  analogous  to  the 
cube  pictured  on  a  plane  surface,  for  the  space  to 
which  we  (being  limited  by  our  tridimensional 
space-conception)  would  naturally  relegate  the  seven 
additional  mirrored  images  is  unoccupied,  and  if 
we  should  make  the  trial,  we  would  find  it  empty. 

Further  experimenting  in  this  line  would  render 
constructions  of  a  more  complicated  character  more 
and  more  difficult  although  not  quite  impossible. 
Thus  we  might  represent  the  formula  (a+^)'*  by 
placing  a  wire  model  of  a  cube,  representing  the 
proportions  (a^b)'\  in  the  corner  of  our  three  mir- 
rors, and  we  would  then  verify  by  ocular  inspec- 
tion the  truth  of  the  formula 

a'+4a'b+6a^b^+4ab^+b\ 

However,  we  must  bear  in  mind  that  all  the 
solids  here  seen  are  merely  the  boundaries  of  four- 


MATHEMATICS   AND   METAGEOMETin'.  95 

dimensional  bodies.  All  of  them  with  the  exception 
of  the  ones  in  the  inner  corner  are  scattered  around 
and  yet  the  analogous  figures  would  have  to  be  re- 
garded as  being  most  intimately  interconnected, 
each  set  of  them  forming  one  four-dimensional  com- 
plex. Their  separation  is  in  appearance  only,  being 
due  to  the  insufficiency  of  our  method  of  presenta- 
tion. 

^\'e  might  obviate  this  fault  by  parceling  our 
wire  cube  and  instead  of  using  three  large  mirrors 
for  reflecting  the  entire  cube  at  once,  we  might  in- 
sert in  its  dividing  planes  double  mirrors,  i.  e.,  mir- 
rors which  would  reflect  on  the  one  side  the  magni- 
tude a  and  on  the  other  the  magnitude  b.  In  this 
way  we  would  come  somewhat  nearer  to  a  faithful 
representation  of  the  nature  of  four-dimensional 
space,  but  the  model  l)eing  divided  up  into  a  number 
of  mirror-walled  rooms,  would  become  extremely 
com])licatcd  and  it  would  be  difficult  for  us  to  bear 
always  in  mind  that  the  mirrored  spaces  count  on 
both  sides  at  once,  although  they  overlap  and  ( tri- 
dimensional] v  considered)  seem  to  fall  the  one  into 
the  other,  thus  ])resenting  to  our  eyes  a  real  laby- 
rinth of  spaces  that  exist  within  each  other  without 
interfering  with  one  another.  Hicy  thus  render 
lU'w  dc])llis  \i^il)]c  in  all  llircc  diniensions,  and  in 
order  to  represent  the  whole  scheme  of  a  four-dimrn 
sional  complex  in  its  full  completeness,  we  ought 
to  have  three  nn'rrors  at  right  angles  placed  at 
cverv  i)oint  in  r)nr  Iridimcnsional  si)ace.  The  scheme 
itself   is   imj)Ossib1e,   but    tlu-   idea    will    render   tlu^ 


96  THE   FOUNDATIONS  OF   MATHEMATICS. 

nature  of  four-dimensional  space  approximately 
clear.  If  we  were  four-dimensional  beings  we  would 
be  possessed  of  the  mirror-eye  which  in  every  di- 
rection could  look  straightway  round  every  corner 
of  the  third  dimension.  This  seems  incredible,  but  it 
can  not  be  denied  that  tridimensional  space  lies 
open  to  an  inspection  from  the  domain  of  the  fourth 
dimension,  just  as  every  point  of  a  Euclidean  plane 
is  open  to  inspection  from  above  to  tridimensional 
vision.  Of  course  we  may  demur  (as  we  actually 
do)  to  believing  in  the  reality  of  a  space  of  four 
dimensions,  but  that  being  granted,  the  inferences 
can  not  be  doubted. 


THE  APPARENT  ARBITRARINESS  OF  THE  A  PRIORI. 

Since  Riemann  has  generalized  the  conception  of 
space,  the  tridimensionality  of  space  seems  very 
arbitrary. 

Why  are  three  co-ordinates  sufficient  for  pure 
space  determinations?  The  obvious  answer  is,  Be- 
cause we  have  three  planes  in  our  construction  of 
space-boundaries.  We  might  as  well  ask,  why  do 
the  three  planes  cut  the  entire  space  into  8  equal 
parts?  The  simple  answer  is  that  we  have  halved 
space  three  times,  and  2^=8.  The  reason  is  prac- 
tically the  same  as  that  for  the  simpler  question, 
why  have  we  two  halves  if  we  di^ade  an  apple  into 
two  equal  parts? 

These  answers  are  simple  enough,  but  there  is 
another  aspect  of  the  question  which  here  seems  in 


MATHEMATICS  AXD   METAGEOMETRV.  97 

order:  Why  not  continue  the  method  of  halving? 
And  there  is  no  other  answer  than  that  it  is  impos- 
sible. The  two  superadded  planes  Po  and  P3  both 
standing  at  right  angles  to  the  original  plane,  neces- 
sarily halve  each  other,  and  thus  the  four  right  an- 
gles of  each  plane  P2  and  P3  on  the  center  of  inter- 
section, form  a  complete  plane  for  the  same  reason 
that  four  quarters  are  one  whole.  A\'e  have  in  each 
case  four  quarters,  and  */^=i. 

Purely  logical  arguments  (i.  e.,  all  modes  of 
reasoning  that  are  rigorously  a  priori,  the  a  priori 
of  abstract  being)  break  down  and  we  must  resort  to 
the  methods  of  the  a  priori  of  doing.  W'e  cannot 
understand  or  grant  the  argument  without  admit- 
ting the  conception  of  space,  previously  created  by 
a  spread  of  pure  motion.  Kant  would  say  that  we 
need  here  the  data  of  rciue  Anschauiing,  and  Kant's 
reine  Anschaiiung  is  a  product  of  our  motility.  As 
soon  as  we  admit  that  there  is  an  a  priori  of  doing 
(of  free  motility)  and  that  our  conception  of  pure 
space  and  time  (Kant's  rciiic  AnscJiaituug)  is  its 
product,  we  understand  that  our  mathematical  con- 
ceptions cannot  be  derived  from  pure  logic  alone 
but  must  filially  dei)end  upon  our  motility,  viz.,  the 
function  that  begets  our  notion  of  space. 

If  \\e  divide  an  a))])le  by  a  vertical  cut  llirough 
its  center,  we  have  two  halves.  Tf  wc  cut  it  again 
by  a  horizontal  cut  through  its  center,  we  have 
four  quarters.  Tf  we  cut  it  again  with  a  cut  that 
is  at  right  angles  to  both  ])rior  ruts,  we  have  eight 
eightlis.     Tt  is  o1)vif)Us1y  impossible  to  insert  among 


98  THE   FOUNDATIONS   OF   MATHEMATICS. 

these  three  cuts  a  fourth  cut  that  would  stand  at 
right  angles  to  these  others.  The  fourth  cut  through 
the  center,  if  we  needs  would  have  to  make  it,  will 
fall  into  one  of  the  prior  cuts  and  be  a  mere  repeti- 
tion of  it,  producing  no  new  result;  or  if  we  made 
it  slanting,  it  would  not  cut  all  eight  parts  but  only 
four  of  them;  it  would  not  produce  sixteen  equal 
parts,  but  twelve  unequal  parts,  viz.,  eight  six- 
teenths plus  four  quarters. 

If  we  do  not  resort  to  a  contemplation  of  the 
scope  of  motion,  if  we  neglect  to  represent  in  our 
imagination  the  figure  of  the  three  planes  and  rely 
on  pure  reason  alone  (i.  e.,  the  rigid  a  priori),  we 
have  no  means  of  refuting  the  assumption  that  we 
ought  to  be  able  to  continue  halving  the  planes  by 
other  planes  at  right  angles.  Yet  is  the  proposition 
as  inconsistent  as  to  expect  that  there  should  be 
regular  pentagons,  or  hexagons,  or  triangles,  the 
angles  of  which  are  all  ninety  degrees. 

From  the  standpoint  of  pure  reason  alone  we 
cannot  disprove  the  incompatibility  of  the  idea  of  a 
rectangular  pentagon.  If  we  insist  on  constructing 
by  hook  or  crook  a  rectangular  pentagon,  we  will 
succeed,  but  we  must  break  away  from  the  straight 
line  or  the  plane.  A  rectangular  pentagon  is  not 
absolutely  impossible;  it  is  absolutely  impossible  in 
the  plane ;  and  if  we  produce  one,  it  will  be  twisted. 

Such  was  the  result  of  Lobatchevsky's  and  Bol- 
yai's  construction  of  a  system  of  geometry  in  which 
the  theorem  of  parallels  does  not  hold.  Their  ge- 
ometries cease  to  be  even ;  they  are  no  longer  Euclid- 


MATHEMATICS  AND   METAGEOMETRY.  QQ 

ean  and  render  the  txtn  boundary  conceptions  un- 
available as  standards  of  measurement. 

If  by  logic  we  understand  consistency,  and  if 
anything  that  is  self-contradictory  and  incompatible 
with  its  own  nature  be  called  illogical,  we  would 
say  that  it  is  not  the  logic  of  pure  reason  that  ren- 
ders certain  things  impossible  in  our  geometric  con- 
structions, but  the  logic  of  our  scope  of  motion.  The 
latter  introduces  a  factor  which  determines  the  na- 
ture of  geometry,  and  if  this  factor  is  neglected  or 
misunderstood,  the  fundamental  notions  of  geom- 
etry must  appear  arbitrary. 

DEFINITENESS    OF    CONSTRUCTION. 

The  problems  which  puzzle  some  of  the  meta- 
physicians of  geometry  seem  to  have  one  common 
foundation,  which  is  the  definiteness  of  geometrical 
construction.  Geometry  starts  from  empty  nothing- 
ness, and  we  are  confronted  with  rigid  conditions 
which  it  does  not  lie  in  our  power  to  change.  We 
make  a  construction,  and  the  result  is  something 
new,  perhaps  something  which  we  have  not  in- 
tended, something  at  which  we  are  surprised.  The 
synthesis  is  a  product  of  our  own  making,  yet  there 
is  an  objective  clement  in  it  nvvv  which  wc  have  no 
command,  and  this  objcctivo  clement  is  rigid,  un- 
compromising, an  irrefragable  necessity,  a  stubborn 
fact,  immm-able,  inflexible,  immutable.    What  is  it? 

Our  metageometricians  overlnf)k  the  fact  that 
their  nothingness  is  not   an  al)solute  nothing,  but 


lOO         THE   FOUNDATIONS   OF   MATHEMATICS. 

only  an  absence  of  concreteness.  If  they  make  defi- 
nite constructions,  they  must  (if  they  only  remain 
consistent)  expect  definite  results.  This  definite- 
ness  is  the  logic  that  dominates  their  operations. 
Sometimes  the  results  seem  arbitrary,  but  they  never 
are ;  for  they  are  necessary,  and  all  questions  why? 
can  elicit  only  answers  that  turn  in  a  circle  and  are 
mere  tautologies. 

Why,  we  may  ask,  do  two  straight  lines,  if  they 
intersect,  produce  four  angles?  Perhaps  we  did 
not  mean  to  construct  angles,  but  here  we  have 
them  in  spite  of  ourselves. 

And  w^hy  is  the  sum  of  these  four  angles  equal 
to  360°  ?  Why,  if  two  are  acute,  will  the  other  two 
be  found  obtuse?  Why,  if  one  angle  be  a  right 
angle,  will  all  four  be  right  angles?  Why  will  the 
sum  of  any  two  adjacent  angles  be  equal  to  two 
right  angles?  etc.  Perhaps  we  should  have  pre- 
ferred three  angles  only,  or  four  acute  angles,  but 
we  cannot  have  them,  at  least  not  by  this  construc- 
tion. 

We  have  seen  that  the  tridimensionality  of  space 
is  arbitrary  only  if  we  judge  of  it  as  a  notion  of 
pure  reason,  without  taking  into  consideration  the 
method  of  its  construction  as  a  scope  of  mobility, 
Tridimensionality  is  only  one  instance  of  apparent 
arbitrariness  among  many  others  of  the  same  kind. 

We  cannot  enclose  a  space  in  a  plane  by  any 
figure  of  two  straight  lines,  and  we  cannot  construct 
a  solid  of  three  even  surfaces. 

There  are  only  definite  forms  of  polyhedra  pos- 


MATHEMATICS  AXU   METAGEOMETRY.  lOI 

sible,  and  the  surfaces  of  every  one  are  definitely 
determined.  To  the  mind  iminitiated  into  the  se- 
crets of  mathematics  it  would  seem  arbitrary  that 
there  are  two  hexahedra  (viz.,  the  cube  and  the 
duplicated  tetrahedron),  while  there  is  no  hepta- 
hedron.  And  why  can  we  not  have  an  octahedron 
with  quadrilateral  surfaces  ?  We  might  as  well  ask, 
why  is  the  square  not  round ! 

Prof.  G.  B.  Halsted  says  in  the  Translator's 
Appendix  to  his  Eng-lish  edition  of  Lobatchevsky's 
Theory  of  Parallels,  p.  48: 

"But  is  it  not  absurd  to  speak  of  space  as  interfering 
with  anything?  If  you  think  so,  take  a  knife  and  a  raw 
potato  and  try  to  cut  it  into  a  seven-edged  solid." 

Truly  Professor  Halsted's  contention,  that  the 
laws  of  space  interfere  with  our  operations,  is  true. 
Yet  it  is  not  space  that  squeezes  us,  but  the  laws  of 
construction  determine  the  shape  of  the  figures 
which  we  make. 

A  simple  instance  that  illustrates  the  way  in 
which  space  interferes  with  our  plans  and  move- 
ments is  the  impossible  demand  on  the  chessboard 
to  start  a  rook  in  one  corner  (Ai)  and  pass  it  with 
the  rook  motion  over  all  the  fields  nncc.  1)nt  only 
once,  and  let  it  end  its  journey  on  the  ojiposiie  corner 
(H8).  Rightly  considered  it  is  not  space  that  inter- 
feres with  our  mode  of  action,  but  the  law  of  con- 
sistency. The  proposition  does  not  contain  any- 
thing: illogical ;  the  words  arc  fjuite  rational  and  the 
sentences  cframmaticallv  correct,     ^'et  is  the  task 


102 


THE    FOUNDATIONS   OF   MATHEMATICS. 


impossible,  because  we  cannot  turn  to  the  right  and 
left  at  once,  nor  can  we  be  in  two  places  at  once, 
neither  can  we  undo  an  act  once  done  or  for  the 

ABCDEFGH 


ABCDEFGH 

nonce  change  the  rook  into  a  bishop ;  but  something 
of  that  kind  would  have  to  be  done,  if  we  start  from 
A I   and  pass  with  the  rook  motion  through  A2 
and  Bi  over  to  B2.  In  other  words: 
Though  the  demand  is  not  in  con- 
flict with  the  logic  of  abstract  being 
or  the  grammar  of  thinking,  it  is 
impossible  because  it  collides  with 
the  logic  of  doing ;  the  logic  of  mov- 
ing about,  the  a  priori  of  motility. 
The  famous  problem  of  crossing  seven  bridges 
leading  to  the  two  Konigsberg  Isles,  is  of  the  same 


MATHEMATICS  AND   METAGEOMETRY. 


103 


kind.  Near  the  mouth  of  the  Pregel  River  there 
is  an  island  called  Kneiphof,  and  the  situation  of 
the  seven  bridges  is  shown  as  in  the  adjoined  dia- 


THE   SEVEN   BRIDGES   OF    KONIGSBERG. 


gram.    A  discussion  arose  as  to  whether  it  was  pos- 
sible to  cross  all  the  bridges  in  a  single  promenade 


EULER  S   DIAGRAM. 


without  crossing  any  one  a  second  time.     Finally 
Euler  solved  the  problem  in  a  memoir  presented  to 


I04        THE   FOUNDATIOXS   OF    MATHEMATICS. 

the  Academy  of  Sciences  of  St.  Petersburg  in  1736, 
pointing-  out  why  the  task  could  not  be  done.  He 
reproduced  the  situation  in  a  diagram  and  proved 
that  if  the  number  of  hues  meeting  at  the  point  K 
(representing  the  island  Kneiphof  as  K)  were  even 
the  task  was  possible,  but  if  the  number  is  odd  it 
can  not  be  accomplished. 

The  sciuaring  of  the  circle  is  similarly  an  im- 
possibility. 

We  cannot  venture  on  self-contradictory  enter- 
prises without  being  defeated,  and  if  the  relation 
of  the  circumference  to  the  diameter  is  an  infinite 
transcendent  scries,  being 

4"  —  J- —  3-rs  —  7-1-9  —  ii-ri3  —  is-ri7  —  i9-r2i  —  23-r'**' 

we  cannot  expect  to  square  the  circle. 

If  we  compute  the  series,  ir  becomes  3. 141 59265 
.  . .  . ,  figures  which  seem  as  arbitrary  as  the  most 
whimsical  fancy. 

It  does  not  seem  less  strange  that  6=2.71828; 
and  yet  it  is  as  little  arbitrary  as  the  ec[uation  3X4 
=12. 

The  definiteness  of  our  mathematical  construc- 
tions and  arithmetical  computations  is  based  upon 
the  inexorable  law  of  determinism,  and  everything 
is  fixed  bv  the  mode  of  its  construction. 


ONE  SPACE,  BUT  VARIOUS  SYSTEMS  OF  SPACE- 
MEASUREMENT. 

Riemann  has  generalized  the  idea  of  space  and 
would  tlius  justify  us  in  speaking  of  ''spaces."   The 


MATHEMATICS   AND   METAGEOMKTRY.  IO5 

common  notion  of  space,  which  agrees  best  with 
that  of  EucHdean  geometry,  has  been  degraded 
into  a  mere  species  of  space,  one  possible  instance 
among  many  other  possibilities.  And  its  very  legit- 
imacy has  been  doubted,  for  it  has  come  to  be  looked 
upon  in  some  quarters  as  only  a  popular  (not  to 
say  vulgar  and  commonplace)  notion,  a  mere  work- 
ing hypothesis,  infested  with  many  arbitrary  con- 
ditions of  which  the  ideal  conception  of  absolute 
space  should  be  free.  How  much  more  interesting 
and  aristocratic  are  curved  space,  the  dainty  two- 
dimensional  space,  and  above  all  the  four-dimen- 
sional space  with  its  magic  powers ! 

The  new  space-conception  seems  bewildering. 
Some  of  these  new  spaces  are  constructions  that  are 
not  concretely  reprcsentable,  but  only  abstractly 
thinkable ;  yet  they  allow  us  to  indulge  in  ingenious 
dreams.  Think  only  of  two-dimensional  creatures, 
and  how  limited  they  are!  They  can  have  no  con- 
ception of  a  third  dimension !  Then  think  of  four- 
dimensional  beings ;  how  superior  they  must  be  to 
us  poor  tridimensional  bodies!  As  we  can  take  a 
figure  situated  within  a  circle  through  the  third  di- 
mension and  put  it  down  again  outside  the  circle 
without  crossing  the  circumference,  so  four-dimen- 
sional beings  could  take  tridinu'iisional  things  en- 
cased in  a  tridimensional  box  from  llieir  hiding- 
place  and  put  them  back  on  some  other  spot  on  the 
outside.  They  could  easily  hel])  themselves  to  all 
the  money  in  the  steal-lined  safes  of  our  banks,  and 
they  could   perform   the   most   difiicult   obstetrical 


I06        THE   FOUNDATIONS   OF   MATHEMATICS. 

feats  without  resorting  to  the  dangerous  Csesarean 
operation. 

Curved  space  is  not  less  interesting.  Just  as 
Hght  may  pass  through  a  medium  that  offers  such 
a  resistance  as  will  involve  a  continuous  displace- 
ment of  the  rays,  so  in  curved  space  the  lines  of 
greatest  intensity  would  be  subject  to  a  continuous 
modification.  The  beings  of  curved  space  may  be 
assumed  to  have  no  conception  of  truly  straight 
lines.  They  must  deem  it  quite  natural  that  if  they 
walk  on  in  the  straightest  possible  manner  they  will 
finally  but  unfailingly  come  back  to  the  same  spot. 
Their  world-space  is  not  as  vague  and  mystical  as 
ours:  it  is  not  infinite,  hazy  at  a  distance,  vague 
and  without  end,  but  definite,  well  rounded  ofif,  and 
perfect.  Presumably  their  lives  have  the  same  ad- 
vantages moving  in  boundless  circles,  while  our 
progression  in  straight  lines  hangs  between  two 
infinitudes — the  past  and  the  future! 

All  these  considerations  are  very  interesting  be- 
cause they  open  new  vistas  to  imaginative  specula- 
tors and  inventors,  and  we  cannot  deny  that  the 
generalization  of  our  space-conception  has  proved 
helpful  by  throwing  new  light  upon  geometrical 
problems  and  widening  the  horizon  of  our  mathe- 
matical knowledge. 

Nevertheless  after  a  mature  deliberation  of  Rie- 
mann's  proposition,  I  have  come  to  the  conclusion 
that  it  leads  us  off  in  a  wrong  direction,  and  in 
contrast  to  his  conception  of  space  as  being  one 
instance  among  many  possibilities,  I   would  insist 


MATHEMATICS  AND   METAGEOMETRV.  lOJ 

Upon  the  uniqueness  of  space.  Space  is  the  possi- 
bihty  of  motion  in  all  directions,  and  mathematical 
space  is  the  ideal  construction  of  our  scope  of  motion 
in  all  directions. 

The  homogeneity  of  space  is  due  to  our  abstrac- 
tion which  omits  all  particularities,  and  its  homa- 
loidality  means  only  that  straight  lines  are  possible 
not  in  the  real  world,  but  in  mathematical  thought, 
and  will  serve  us  as  standards  of  measurement. 

Curved  space,  so  called,  is  a  more  complicated 
construction  of  space-measurement  to  which  some 
additional  feature  of  a  particular  nature  has  been 
admitted,  and  in  which  we  waive  the  advantages  of 
even  boundaries  as  means  of  measurement. 

Space,  the  actual  scope  of  motion,  remains  dif- 
ferent from  all  systems  of  space-measurement,  be 
they  homaloidal  or  curved,  and  should  not  be  sub- 
sumed with  them  under  one  and  the  same  category. 

Riemann's  several  space  -  conceptions  are  not 
spaces  in  the  proper  sense  of  the  word,  but  systems 
of  space-measurement.  Tt  is  true  that  space  is  a 
tridimensional  mam' fold,  and  a  ])]anc  a  two-dimen- 
sional manifold,  and  we  can  think  of  other  systems 
of  ;/-manifoldncss ;  but  for  that  reason  all  these  dif- 
ferent manifr)lds  do  not  become  spaces.  Man  is  a 
mammal  having  two  prehensiles  (his  hands)  ;  the 
elephant  is  a  mammal  with  one  prehensile  (hi-^ 
trunk)  ;  tailless  monkeys  like  the  pavian  have  four; 
and  tailed  monkeys  have  five  prehensiles.  Ts  there 
any  logic  in  extending  the  dcnonu'nation  man  to  all 
these  animals,  and  should  we  define  the  clcj)hant 


I08        THE   FOUNDATIONS   OF   MATHEMATICS. 

as  a  man  with  one  prehensile,  the  pavian  as  a  man 
with  four  prehensiles  and  tailed  monkeys  as  men 
with  five  prehensiles  ?  Our  zoolog"ists  would  at  once 
protest  and  denounce  it  as  an  illogical  misuse  of 
names. 

Space  is  a  manifold,  but  not  every  manifold  is 
a  space. 

Of  course  every  one  has  a  right  to  define  the 
terms  he  uses,  and  obviously  my  protest  simply  re- 
jects Riemann's  use  of  a  word,  but  I  claim  that  his 
identification  of  "space"  with  "manifold"  is  the 
source  of  inextricable  confusion. 

It  is  well  known  that  all  colors  can  be  reduced 
to  three  primary  colors,  yellow,  red,  and  blue,  and 
thus  we  can  determine  any  possible  tint  by  three 
co-ordinates,  and  color  just  as  much  as  mathemat- 
ical space  is  a  threefold,  viz.,  a  system  in  which 
three  co-ordinates  are  needed  for  the  determination 
of  any  thing.  But  because  color  is  a  threefold,  no 
one  would  assume  that  color  is  space. 

Riemann's  manifolds  are  systems  of  measure- 
ment, and  the  system  of  three  co-ordinates  on 
three  intersecting  planes  is  an  a  priori  or  purely 
formal  and  ideal  construction  invented  to  calculate 
space.  We  can  invent  other  more  complicated  sys- 
tems of  measurement,  with  curved  lines  and  with 
more  than  three  or  less  than  three  co-ordinates. 
We  can  even  employ  them  for  space-measurement, 
although  they  are  rather  awkward  and  unservice- 
able; but  these  systems  of  measurement  are  not 
"spaces,"  and  if  they  are  called  so,  they  are  spaces 


MATHEMATICS  AND   METAGEOMETRV.  IO9 

by  courtesy  only.  By  a  metaphorical  extension  we 
allow  the  idea  of  system  of  space-measurement  to 
stand  for  space  itself.  It  is  a  brilliant  idea  and  quite 
as  ingenious  as  the  invention  of  animal  fables  in 
^^•hich  our  quadruped  fellow-beings  are  endowed 
with  speech  and  treated  as  human  beings.  But  such 
poetical  licences,  in  which  facts  are  stretched  and 
the  meaning  of  terms  is  slightly  modified,  is  possible 
only  if  instead  of  the  old-fashioned  straight  rules  of 
logic  we  grant  a  slight  curvature  to  our  syllogisms. 

FICTITIOUS    SPACES    AND    THE    APRIORITY    OF    ALL 
SYSTEMS    OF    SPACE-MEASUREMENT. 

Mathematical  space,  so  called,  is  strictly  speak- 
ing no  space  at  all,  but  the  mental  construction  of 
a  manifold,  being  a  tridimensional  system  of  space- 
measurement  invented  for  the  determination  of  ac- 
tual space. 

Neither  can  a  manifold  of  two  dimensions  be 
called  a  space.  It  is  a  mere  Ixtundarv  in  space,  it 
is  no  reality,  but  a  conce])t.  a  construction  of  ])ure 
thought. 

Furtlier,  the  manifold  of  four  dimensions  is  a 
system  ot  iiu-a^iirc'iiK-nl  applicable  to  ;in\'  re.alitv 
for  the  detcnnination  of  which  four  coordinates 
are  needed.  It  is  applicable  to  re;il  space  if  there 
is  connected  with  it  in  addition  to  the  three  planes 
at  right  angles  another  condition  of  a  constant  na- 
ture, such  as  gravity. 

At  any  rale,  we  niU'-l  den\-  the  applicabilil  \'  of  a 


no        THE   FOUNDATIONS  OF   MATHEMATICS. 

system  of  four  dimensions  to  empty  space  void  of 
any  such  particularity.  The  idea  of  space  being 
four-dimensional  is  chimerical  if  the  word  space  is 
used  in  the  common  acceptance  of  the  term  as 
juxtaposition  or  as  the  scope  of  motion.  So  long- 
as  four  quarters  make  one  whole,  and  four  right 
angles  make  one  entire  circumference,  and  so  long 
as  the  contents  of  a  sphere  which  covers  the  entire 
scope  of  motion  round  its  center  equals  iirr^,  there 
is  no  sense  in  entertaining  the  idea  that  empty  space 
might  be  four-dimensional. 

But  the  argument  is  made  and  sustained  by 
Helmholtz  that  as  two-dimensional  beings  perceive 
two  dimensions  only  and  are  unable  to  think  how 
a  third  dimension  is  at  all  possible,  so  we  tridimen- 
sional beings  cannot  represent  in  thought  the  possi- 
bility of  a  fourth  dimension.  Helmholtz,  speaking 
of  beings  of  only  two  dimensions  living  on  the  sur- 
face of  a  solid  body,  says: 

"If  such  beings  worked  out  a  geometry,  they  would  of 
course  assign  only  two  dimensions  to  their  space.  They 
would  ascertain  that  a  point  in  moving  describes  a  line,  and 
that  a  line  in  moving  describes  a  surface.  But  they  could 
as  little  represent  to  themselves  what  further  spatial  con- 
struction would  be  generated  by  a  surface  moving  out  of 
itself,  as  we  can  represent  what  should  be  generated  by  a 
solid  moving  out  of  the  space  we  know.  By  the  much- 
abused  expression  'to  represent'  or  'to  be  able  to  think  how 
something  happens'  I  understand — and  I  do  not  see  how 
anything  else  can  be  understood  by  it  without  loss  of  all 
meaning — the  power  of  imagining  the  whole  series  of  sen- 
sible impressions  that  would  be  had  in  such  a  case.     Now, 


MATHEMATICS  AND   METAGEOMETRY.  Ill 

as  no  sensible  impression  is  known  relating  to  such  an  un- 
heard-of event,  as  the  movement  to  a  fourth  dimension 
would  be  to  us,  or  as  a  movement  to  our  third  dimension 
would  be  to  the  inhabitants  of  a  surface,  such  a  'represen- 
tation' is  as  impossible  as  the  'representation'  of  colors 
would  be  to  one  born  blind,  if  a  description  of  them  in  gen- 
eral terms  could  be  given  to  him. 

"Our  surface-beings  would  also  be  able  to  draw  shortest 
lines  in  their  superficial  space.  These  would  not  necessarily 
be  straight  lines  in  our  sense,  but  what  are  technically  called 
geodetic  lines  of  the  surface  on  which  they  live ;  lines  such 
as  are  described  by  a  tense  thread  laid  along  the  surface, 
and  which  can  slide  upon  it  freely.".  . .  . 

"Now,  if  beings  of  this  kind  lived  on  an  infinite  plane, 
their  geometry  would  be  exactly  the  same  as  our  planimetry. 
They  would  affirm  that  only  one  straight  line  is  possible 
between  two  points ;  that  through  a  third  point  lying  with- 
out this  line  only  one  line  can  be  drawn  parallel  to  it ;  that 
the  ends  of  a  straight  line  never  meet  though  it  is  produced 
to  infinity,  and  so  on.".  . .  . 

"But  intelligent  beings  of  the  kind  supposed  might  also 
live  on  the  surface  of  a  sjihcre.  Their  shortest  or  straightest 
line  between  two  points  would  then  be  an  arc  of  the  great 
circle  passing  through  them.".  . .  . 

"Of  parallel  lines  the  sphere-dwellers  would  know  noth- 
ing. They  would  maintain  that  any  two  straightest  lines, 
sufficiently  produced,  must  finally  cut  not  in  one  only  but  in 
two  points.  The  sum  of  the  angles  of  a  triangle  would  be 
always  greater  than  two  right  angles,  increasing  as  the  sur- 
face of  the  triangle  grew  greater.  They  could  thus  have 
no  conception  of  geometrical  similarity  between  greater  and 
smaller  figures  of  the  same  kind,  for  with  them  a  greater 
triangle  must  have  different  angles  from  a  smaller  one. 
Their  space  would  be  unlimited,  but  wouUl  be  foiuifl  to  be 
finite  or  at  least  represented  as  such. 

"Tt  is  clear,  then,  that  such  beings  uuist  set  up  a  very 


112         THE   FOUNDATIONS   OF    MATHEMATICS. 

different  system  of  geometrical  axioms  from  that  of  the 
inhabitants  of  a  plane,  or  from  ours  with  our  space  of  three 
dimensions,  though  the  logical  powers  of  all  were  the  same." 

I  deny  what  Helmholtz  implicitly  assumes  that 
sensible  impressions  enter  into  the  fabric  of  our 
concepts  of  purely  formal  relations.  We  have  the 
idea  of  a  surface  as  a  boundary  between  solids,  but 
surfaces  do  not  exist  in  reality.  All  real  objects 
are  solid,  and  our  idea  of  surface  is  a  mere  fiction 
of  abstract  reasoning.  Two  dimensional  things  are 
unreal,  w^e  have  never  seen  any,  and  yet  we  form 
the  notion  of  surfaces,  and  lines,  and  points,  and 
pure  space,  etc.  There  is  no  straight  line  in  ex- 
istence, hence  it  can  produce  no  sense-impression, 
and  yet  we  ha^'e  the  notion  of  a  straight  line.  The 
straight  lines  on  paper  are  incorrect  pictures  of  the 
true  straight  lines  which  are  purely  ideal  construc- 
tions. Our  a  priori  constructions  are  not  a  product 
of  our  sense-impressions,  but  are  independent  of 
sense  or  anything  sensed. 

It  is  of  course  to  be  granted  that  in  order  to 
have  any  conception,  we  must  have  first  of  all  sen- 
sation, and  we  can  gain  an  idea  of  pure  form  only 
by  abstraction.  But  having  gained  a  fund  of  ab- 
stract notions,  we  can  generalize  them  and  modify 
them;  we  can  use  them  as  a  child  uses  its  building 
blocks,  we  can  make  constructions  of  pure  thought 
unrealizable  in  the  concrete  world  of  actuality.  Some 
of  such  constructions  cannot  be  represented  in  con- 
crete form,  but  they  are  not  for  that  reason  un- 
thinkable.    Even  if  we  grant  that  two-dimensional 


MATHEMATICS  AND   METAGEOMETRY.  IT^ 

beings  were  possible,  we  would  have  no  reason  to 
assume  that  two-dimensional  beings  could  not  con- 
struct a  tridimensional  space-conception. 

Two-dimensional  beings  could  not  be  possessed 
of  a  material  body,  because  their  absolute  flatness 
substantially  reduces  their  shape  to  nothingness. 
But  if  they  existed,  they  would  be  limited  to  move- 
ments in  two  directions  and  thus  must  be  expected 
to  be  incredulous  as  to  the  possibility  of  jumping 
out  of  their  flat  existence  and  returning  into  it 
through  a  third  dimension.  Having  never  moved 
in  a  third  dimension,  they  could  speak  of  it  as  the 
blind  might  discuss  colors ;  in  their  flat  minds  they 
could  have  no  true  conception  of  its  significance  and 
would  be  unable  to  clearly  picture  it  in  their  imagi- 
nation; but  for  all  their  limitations,  they  could  very 
well  develop  the  abstract  idea  of  tridimensional 
space  and  therefrom  derive  all  particulars  of  its 
laws  and  conditions  and  possibilities  in  a  similar 
way  as  we  can  acf|uire  the  notion  of  a  space  of  four 
dimensions. 

Hclmholtz  continues: 

"Rut  let  ns  proceed  still  farther. 

"Let  us  think  of  reasoning-  beinj^s  existing;  on  the  surface 
of  an  eg^ff-shaped  body.  Shortest  lines  could  be  drawn  be- 
tween three  points  of  such  a  surface  and  a  triangle  con- 
structed. Rut  if  the  attempt  were  made  to  construct  con- 
Srruent  triang^les  at  different  parts  of  the  surface,  it  would 
be  found  that  two  triangles,  with  three  j)airs  of  equal  sides, 
would  not  have  their  ang-les  equal." 

Tf  there  were  two-dimensional  bcinLr^  liv-ino-  on 


114        THE   FOUNDATIONS   OF   MATHEMATICS. 

an  egg-shell,  they  would  most  likely  have  to  deter- 
mine the  place  of  their  habitat  by  experience  just 
as  much  as  we  tridimensional  beings  living  on  a 
flattened  sphere  have  to  map  out  our  world  by  meas- 
urements made  a  posteriori  and  based  upon  a  priori 
systems  of  measurement. 

If  the  several  systems  of  space-measurement 
were  not  a  priori  constructions,  how  could  Helm- 
holtz  who  does  not  belong  to  the  class  of  two-dimen- 
sional beings  tell  us  what  their  notions  must  be 
like? 

I  claim  that  if  there  were  surface  beings  on  a 
sphere  or  on  an  egg-shell,  they  would  have  the 
same  a  priori  notions  as  we  have;  they  would  be 
able  to  construct  straight  lines,  even  though  they 
were  constrained  to  move  in  curves  only ;  they  would 
be  able  to  define  the  nature  of  a  space  of  three  di- 
mensions and  would  probably  locate  in  the  third 
dimension  their  gods  and  the  abode  of  spirits.  I 
insist  that  not  sense  experience,  but  a  priori  con- 
siderations, teach  us  the  notions  of  straight  lines. 

The  truth  is  that  we  tridimensional  beings  ac- 
tually do  live  on  a  sphere,  and  we  cannot  get  away 
from  it.  What  is  the  highest  flight  of  an  aeronaut 
and  the  deepest  descent  into  a  mine  if  measured  by 
the  radius  of  the  earth?  If  we  made  an  exact  imi- 
tation of  our  planet,  a  yard  in  diameter,  it  would 
be  like  a  polished  ball,  and  the  highest  elevations 
would  be  less  than  a  grain  of  sand ;  they  would  not 
be  noticeable  were  it  not  for  a  difference  in  color 
and  material. 


MATHEMATICS  AND   METAGEOMETRY.  II5 

\\'hen  we  become  conscious  of  the  nature  of  our 
hal)itation,  we  do  not  construct  a  priori  conceptions 
accordingly,  Init  feel  limited  to  a  narrow  surface 
and  behold  with  wonder  the  infinitude  of  space  be- 
yond. We  can  very  well  construct  other  a  priori 
notions  which  would  be  adapted  to  one,  or  two,  or 
four-dimensional  worlds,  or  to  spaces  of  positive 
or  of  negative  curvatures,  for  all  these  constructions 
are  ideal ;  thev  are  mind-made  and  we  select  from 
them  the  one  that  would  best  serve  our  purpose  of 
space-measurement. 

The  claim  is  made  that  if  we  were  four-dimen- 
sional beings,  our  present  three-dimensional  world 
would  appear  to  us  as  flat  and  shallow,  as  the  plane 
is  to  us  in  our  present  tridimensional  predicament. 
That  statement  is  true,  because  it  is  conditioned  by 
an  "if."  And  what  pretty  romances  have  been 
built  upon  it!  I  remind  my  reader  only  of  the  in- 
genious story  Flatland,  Writ  fen  by  a  Square,  and 
portions  of  Wilhelm  Busch's  charming  tale  Ed- 
zvard's  Dreaur:  but  the  worth  of  conditional  truths 
dci)cnds  ui)on  the  assumptinn  upon  which  they  are 
made  contingent,  and  the  argument  is  easy  enough 
that  if  things  were  different,  they  would  not  be  what 
they  are.  If  T  had  wings,  I  could  Hy ;  if  I  had  gills 
F  could  live  under  water:  if  1  were  a  magician  I 
coulfl  work  nu'racles. 

'  'I'hc  Open  Lourt,  Vol.  VIII,  p.  4266  ct  seq. 


Il6         THE   FOUNDATIONS   OF   MATHEMATICS. 
INFINITUDE. 

The  notion  is  rife  at  present  that  infinitude  is 
self-contradictory  and  impossible.  But  that  notion 
originates  from  the  error  that  space  is  a  thing,  an 
objective  and  concrete  reality,  if  not  actually  mate- 
rial, yet  consisting-  of  some  substance  or  essence. 
It  is  true  that  infinite  things  cannot  exist,  for  things 
are  always  concrete  and  limited;  but  space  is  pure 
potentiality  of  concrete  existence.  Pure  space  is 
'materially  considered  nothing.  That  this  pure  space 
(this  apparent  nothing)  possesses  some  very  defi- 
nite positive  qualities  is  a  truth  which  at  first  sight 
may  seem  strange,  but  on  closer  inspection  is  quite 
natural  and  will  be  conceded  by  every  one  who  com- 
prehends the  paramount  significance  of  the  doctrine 
of  pure  form. 

Space  being  pure  form  of  extension,  it  must  be 
infinite,  and  infinite  means  that  however  far  we  go, 
in  whatever  direction  we  choose,  we  can  go  farther, 
and  will  never  reach  an  end.  Time  is  just  as  infinite 
as  space.  Our  sun  will  set  and  the  present  day  will 
pass  away,  but  time  will  not  stop.  We  can  go  back- 
ward to  the  beginning,  and  we  must  ask  what  was 
before  the  beginning.  Yet  suppose  w^e  could  fill  the 
blank  with  some  hypothesis  or  another,  mytholog- 
ical or  metaphysical,  we  would  not  come  to  an  ab- 
solute beginning.  The  same  is  true  as  to  the  end. 
And  if  the  universe  broke  to  pieces,  time  would 
continue,  for  even  the  duration  in  which  the  world 
would  lie  in  ruins  would  be  measurable. 


MATHEMATICS  AND   METAGEOMETRY.  11/ 

Not  only  is  space  as  a  totality  infinite,  but  in 
every  part  of  space  we  have  infinite  directions. 

What  does  it  mean  that  space  has  infinite  direc- 
tions ?  If  you  lay  down  a  direction  by  drawing  a  line 
from  a  g-iven  point,  and  continue  to  lay  down  other 
directions,  there  is  no  way  of  exhausting-  your  pos- 
sibilities. Light  travels  in  all  directions  at  once; 
but  "all  directions"  means  that  the  whole  extent  of 
the  surroundings  of  a  source  of  light  is  agitated, 
and  if  we  attempt  to  gather  in  the  whole  by  picking 
up  every  single  direction  of  it,  we  stand  before  a 
task  that  cannot  be  finished. 

In  the  same  way  any  line,  though  it  be  of  definite 
length,  can  suffer  infinite  division,  and  the  fraction 
'/;{  is  quite  definite  while  the  same  amount  if  ex- 
])ressed  in  decimals  as  0.333 can  never  be  com- 
pleted. Light  actually  travels  in  all  directions, 
which  is  a  definite  and  concrete  process,  but  if  we 
try  to  lay  them  down  one  by  one  we  find  that  we  can 
as  little  cxhruist  tlicir  numl)er  as  we  can  come  to 
an  end  in  di\isi1)ility  or  as  we  can  reacli  the  bound- 
ary of  space,  or  as  we  can  come  to  an  ultimate 
numl)er  in  counting.  In  other  words,  reality  is  ac- 
tual and  definite  1)Ut  our  mode  of  measuring  it  or 
reducing  it  to  fr)rmulas  admits  of  a  more  ov  less  ap- 
proximate treatment  onlv.  being  the-  function  of  an 
infinite  progress  in  some  direction  or  other.  Tlu're 
is  an  r)bjective  raisnn  d'etre  for  ibc  roiirc'i)lion  of 
the  infinite,  but  our  frtrmnlntion  of  it  is  snbji-etive, 
and  the  puzzling  feature  of  it  rtriginates  from  treat- 
ing the  'Subjective  feature  as  an  r)biective  fact. 


Il8        THE  FOUNDATIONS  OF  MATHEMATICS. 

These  considerations  indicate  that  infinitude 
does  not  appertain  to  the  thing,  but  to  our  method 
of  viewing  the  thing.  Things  are  always  concrete 
and  definite,  but  the  relational  of  things  admits  of 
a  progressive  treatment.  Space  is  not  a  thing,  but 
the  relational  feature  of  things.  If  we  say  that 
space  is  infinite,  we  mean  that  a  point  may  move  in- 
cessantly and  will  never  reach  the  end  where  its 
progress  would  be  stopped. 

There  is  a  phrase  current  that  the  finite  cannot 
comprehend  the  infinite.  Man  is  supposed  to  be 
finite,  and  the  infinite  is  identified  with  God  or  the 
Unknowable,  or  anything  that  surpasses  the  com- 
prehension of  the  average  intellect.  The  saying  is 
based  upon  the  prejudicial  conception  of  the  infinite 
as  a  realized  actuality,  while  the  infinite  is  not  a 
concrete  thing,  but  a  series,  a  process,  an  aspect, 
or  the  plan  of  action  that  is  carried  on  without  stop- 
ping and  shall  not,  as  a  matter  of  principle,  be  cut 
short.  Accordingly,  the  infinite  (though  in  its  com- 
pleteness unactualizable)  is  neither  mysterious  nor 
incomprehensible,  and  though  mathematicians  be 
finite,  they  may  very  successfully  employ  the  infinite 
in  their  calculations. 

I  do  not  say  that  the  idea  of  infinitude  presents 
no  difficulties,  but  I  do  deny  that  it  is  a  self-contra- 
dictory notion  and  that  if  space  must  be  conceived 
to  be  infinite,  mathematics  will  sink  into  mysticism. 


MATHEMATICS  AXD   METAGEOMETRY.  I IQ 

GEOMETRY  REMAINS  A  PRIORI. 

Those  of  our  readers  who  have  closely  followed 
our  arguments  will  now  understand  how^  in  one  im- 
portant point  w^e  cannot  accept  Mr.  B.  A.  W.  Rus- 
sell's statement  as  to  the  main  result  of  the  meta- 
geometrical  inquisition.     He  says: 

"There  is  thus  a  complete  divorce  between  geometry  and 
the  study  of  actual  space.  Geometry  does  not  give  us  cer- 
tain knowledge  as  to  what  exists.  That  peculiar  position 
which  geometry  formerly  appeared  to  occupy,  as  an  a  priori 
science  giving  knowledge  of  something  actual,  now  appears 
to  have  been  erroneous.  It  points  out  a  whole  series  of  pos- 
sibilities, each  of  which  contains  a  whole  system  of  con- 
nected propositions ;  but  it  throws  no  more  light  upon  the 
nature  of  our  space  than  arithmetic  throws  upon  the  popu- 
lation of  Great  Britain.  Thus  the  plan  of  attack  suggested 
by  non-Euclidean  geometry  enables  us  to  capture  the  last 
stronghold  of  those  who  attempt,  from  logical  or  a  priori 
considerations,  to  deduce  the  nature  of  what  exists.  The 
conclusion  suggested  is,  that  no  existential  proposition  can 
be  deduced  from  one  which  is  not  existential.  But  to  prove 
such  a  conclusion  would  demand  a  treatise  upon  all  branches 
of  philosophy."^ 

It  is  a  matter  of  course  that  the  single  facts  as  to 
the  poj)ulation  of  Great  Britain  must  be  supplied 
by  counting,  and  in  the  same  way  the  measurements 
of  angles  and  actual  distances  must  be  taken  by  a 
posteriori  transactions;  but  having  ascertained 
some  lines  and  angles,  we  can  (assuming  our  data 
to  be  correct)  calculate  other  items  with  absolute 

'  In  llic  UC7V  vnlumcs  of  Ihc  pjicyclnficrdia  Priinuitica,  Vol. 
XXVIII,  of  the  complete  work,  s.  v.  Geometry,  Non-Euclidean,  p. 
674- 


120        THE   FOUNDATIONS   OF   MATHEMATICS. 

exactness  by  purely  a  priori  argument.  There  is 
no  need  (as  Mr.  Russell  puts  it)  "from  logical  or 
a  priori  considerations  to  deduce  the  nature  of  what 
exists," — which  seems  to  mean,  to  determine  special 
features  of  concrete  instances.  No  one  ever  as- 
sumed that  the  nature  of  particular  cases,  the  c[ual- 
ities  of  material  things,  or  sense-afifecting  proper- 
ties, could  be  determined  by  a  priori  considerations. 
The  real  question  is,  w^hether  or  not  the  theorems 
of  space  relations  and,  generally,  purely  formal  con- 
ceptions, such  as  are  developed  a  priori  in  geom- 
etry and  kindred  formal  sciences,  will  hold  good  in 
actual  experience.  In  other  words,  can  we  assume 
that  form  is  an  objective  quality,  which  would  im- 
ply that  the  constitution  of  the  actual  world  must  be 
the  same  as  the  constitution  of  our  purely  a  priori 
sciences?  ^^>  answer  this  latter  question  in  the 
affirmative. 

We  cannot  determine  by  a  priori  reasoning  the 
population  of  Great  Britain.  But  we  can  a  pos- 
teriori count  the  inhabitants  of  several  towns  and 
districts,  and  determine  the  total  by  addition.  The 
rules  of  addition,  of  division,  and  multiplication  can 
be  relied  upon  for  the  calculation  of  objective  facts. 

Or  to  take  a  geometrical  example.  When  we 
measure  the  distance  between  two  observatories 
and  also  the  angles  at  which  at  either  end  of  the 
line  thus  laid  down  the  moon  appears  in  a  given 
moment,  we  can  calculate  the  moon's  distance  from 
the  earth ;  and  this  is  possible  only  on  the  assump- 
tion that  the  formal  relations  of  objective  space 


MATHEMATICS  AND  METAGEOMETRY.  121 

are  the  same  as  those  of  mathematical  space.  In 
other  words,  that  our  a  priori  mathematical  calcu- 
lations can  be  made  to  throw  light  upon  the  nature 
of  space, — the  real  objective  space  of  the  world  in 
which  we  live. 


The  result  of  our  investigation  is  quite  conserva- 
tive. It  re-establishes  the  apriority  of  mathematical 
space,  yet  in  doing  so  it  justifies  the  method  of  meta- 
physicians in  their  constructions  of  the  several  non- 
Euclidean  systems.  All  geometrical  systems,  Eu- 
clidean as  well  as  non-Euclidean,  are  purely  ideal 
constructions.  If  we  make  one  of  them  we  then 
and  there  for  that  purpose  and  for  the  time  being, 
exclude  the  other  systems,  but  they  are  all,  each 
one  on  its  own  premises,  equally  true  and  the  ques- 
tion of  preference  between  them  is  not  one  of  truth 
or  untruth  but  of  adequacy,  of  practicability,  of  use- 
fulness. 

The  question  is  not:  "Is  real  space  tliat  of  Eu- 
clid or  of  Riemann,  of  Lobatchevsky  or  i>olyai?" 
for  real  space  is  sim])ly  the  juxtapositions  of  things, 
while  our  geometries  are  ideal  schemes,  mental  con- 
structions of  models  for  space  measurement.  The 
real  fjuestion  is,  "Which  system  is  the  most  con- 
venient to  determine  the  juxtaposition  of  things?" 

/I  priori  considered,  all  geometries  have  equal 
rights,  but  fr)r  all  lliat  i-'iiclidean  gennielry,  wbieli 
in  the  parallel  theorem  takes  ilie  bull  by  the  horn, 
will  remain  classical  fr)rever,  for  after  all  the  non- 


122        THE  FOUNDATIONS  OF   MATHEMATICS. 

Euclidean  systems  cannot  avoid  developing  the  no- 
tion of  the  straight  line  or  other  even  boundaries. 
Any  geometry  could,  within  its  own  premises,  be 
utilized  for  a  determination  of  objective  space;  but 
we  will  naturally  give  the  preference  to  plane  ge- 
ometry, not  because  it  is  truer,  but  because  it  is 
simpler  and  will  therefore  be  more  serviceable. 

How  an  ideal  (and  apparently  purely  subjec- 
tive) construction  can  give  us  any  information  of 
the  objective  constitution  of  things,  at  least  so  far  as 
space-relations  are  concerned,  seems  mysterious  but 
the  problem  is  solved  if  we  bear  in  mind  the  objec- 
tive nature  of  the  a  priori, — a  topic  which  we  have 
elsewhere  discussed.* 

SENSE-EXPERIENCE  AND   SPACE. 

We  have  learned  that  sense-experience  cannot 
be  used  as  a  source  from  which  we  construct  our 
fundamental  notions  of  geometry,  yet  sense-experi- 
ence justifies  them. 

Experience  can  verify  a  priori  constructions  as, 
e.  g.,  tridimensionality  is  verified  in  Newton's  laws: 
but  experience  can  never  refute  them,  nor  can  it 
change  them.  We  may  apply  any  system  if  we  only 
remain  consistent.  It  is  quite  indififerent  whether 
we  count  after  the  decimal,  the  binary  or  the  duo- 
decimal system.  The  result  will  be  the  same.  If 
experience  does  not  tally  with  our  calculations,  we 
have  either  made  a  mistake  or  made  a  wrong  ob- 

^  See  also  the  author's  exposition  of  the  problem  of  the  a  priori 
in  his  edition  of  Kant's  Prolegomena,  pp.  167-240. 


MATHEMATICS  AND  METAGEOMETRY.  I23 

servation.  For  our  a  priori  conceptions  hold  good 
for  any  conditions,  and  their  theory  can  l)e  as  httle 
wrong  as  reahty  can  be  inconsistent. 

However,  some  of  the  most  ingenious  thinkers 
and  great  mathematicians  do  not  conceive  of  space 
as  mere  potentiahty  of  existence,  which  renders  it 
formal  and  purely  a  priori,  but  think  of  it  as  a 
concrete  reality,  as  though  it  were  a  big  box,  pre- 
sumably round,  like  an  immeasurable  sphere.  If  it 
were  such,  space  would  be  (as  Riemann  says) 
boundless  but  not  infinite,  for  we  cannot  find  a 
boundary  on  the  surface  of  a  sphere,  and  yet  the 
sphere  has  a  finite  surface  that  can  be  expressed  in 
definite  numbers. 

I  should  like  to  know  what  Riemann  would  call 
that  something  which  lies  outside  of  his  spherical 
space.  Would  the  name  "province  of  the  extra- 
spatial"  perhaps  be  an  appropriate  term?  I  do  not 
know  how  we  can  rid  ourselves  of  this  enormous 
portion  of  unutilized  outside  room.  Strange  though 
it  may  seem,  this  space-conception  of  Riemann 
counts  among  its  advocates  mathematicians  of  first 
rank,  among  whom  I  will  here  mention  only  the 
name  of  Sir  Robert  Rail. 

It  will  be  interesting  to  hear  a  modern  thinker 
who  is  strongly  affected  by  metageometrical  studies, 
on  the  nature  of  space.  Mr.  Charles  S.  Peirce,  an 
uncoiniiioiily  keen  logician  and  an  original  ihinker 
of  no  mean  repute,  j)ropf)ses  the  following  \]]vvc 
alternatixes.     I  le  savs: 


124        THE   FOUNDATIONS   OF   MATHEMATICS. 

"First,  space  is,  as  Euclid  teaches,  both  unlimited  and 
iniineasurable,  so  that  the  infinitely  distant  parts  of  any 
plane  seen  in  perspective  appear  as  a  straight  line,  in  which 
case  the  sum  of  the  three  angles  amounts  to  i8o° ;  or, 

"Second,  space  is  iuinicasurablc  but  limited,  so  that  the 
infinitely  distant  parts  of  any  plane  seen  in  perspective  ap- 
pear as  in  a  circle,  beyond  which  all  is  blackness,  and  in  this 
case  the  sum  of  the  three  angles  of  a  triangle  is  less  than 
i8o°  by  an  amount  proportional  to  the  area  of  the  tri- 
angle ;  or 

"Third,  space  is  unlimited  but  finite  (like  the  surface  of 
a  sphere),  so  that  it  has  no  infinitely  distant  parts;  but  a 
finite  journey  along  any  straight  line  would  bring  one  back 
to  his  original  position,  and  looking  ofif  with  an  unobstructed 
view  one  would  see  the  back  of  his  own  head  enormously 
magnified,  in  which  case  the  sum  of  the  three  angles  of  a 
triangle  exceeds  i8o°  by  an  amount  proportional  to  the 
area. 

"Which  of  these  three  hypotheses  is  true  we  know  not. 
The  largest  triangles  we  can  measure  are  such  as  have  the 
earth's  orbit  for  base,  and  the  distance  of  a  fixed  star  for 
altitude.  The  angular  magnitude  resulting  from  subtracting 
the  sum  of  the  two  angles  at  the  base  of  such  a  triangle 
from  i8o°  is  called  the  star's  parallax.  The  parallaxes  of 
only  about  forty  stars  have  been  measured  as  yet.  Two  of 
them  come  out  negative,  that  of  Arided  (a  Cycni),  a  star 
of  magnitude  lyi,  which  is  — o."o82.  according  to  C.  A.  F, 
Peters,  and  that  of  a  star  of  magnitude  7%,  known  as 
Piazzi  Til  422,  which  is  — ©."045  according  to  R.  S.  Ball. 
But  these  negative  parallaxes  are  undoubtedly  to  be  attrib- 
uted to  errors  of  observation  ;  for  the  probable  error  of 
such  a  determination  is  about  drO."o75,  and  it  would  be 
strange  indeed  if  we  were  to  be  able  to  see,  as  it  were,  more 
than  half  way  round  space,  without  being  able  to  see  stars 
with  larger  negative  parallaxes.  Indeed,  the  very  fact  that 
of  all  the  parallaxes  measured  only  two  come  out  negative 


MATHEMATICS  AXI)   METAGEOMETRY.  125 

would  be  a  strong  argument  that  the  smallest  parallaxes 
really  amount  to  4-o."i,  were  it  not  for  the  reflexion  that 
the  publication  of  other  negative  parallaxes  may  have  been 
suppressed.  I  think  we  may  feel  confident  that  the  parallax 
of  the  furthest  star  lies  somewhere  between  — o."o5  and 
-f-o."i5,  and  within  another  century  our  grandchildren  will 
surely  know  whether  the  three  angles  of  a  triangle  are 
greater  or  less  than  i8o°, — that  they  are  exactly  that  amount 
is  what  nobody  ever  can  be  justified  in  concluding.  Tt  is 
true  that  according  to  the  axioms  of  geometry  the  sum  of 
the  three  sides  of  a  triangle  is  precisely  i8o° ;  but  these 
axioms  are  now  exploded,  and  geometers  confess  that  they, 
as  geometers,  know  not  the  slightest  reason  for  supposing 
them  to  be  precisely  true.  They  are  expressions  of  our  in- 
born conception  of  space,  and  as  such  are  entitled  to  credit, 
so  far  as  their  truth  could  have  influenced  the  formation  of 
the  mind.  But  that  affords  not  the  slightest  reason  for  sup- 
posing them  exact."     (The  Monist,  \'ol.  I.  pp.  173-174.) 

Now,  let  us  for  argument's  sake  assume  that 
the  measurements  of  star-parallaxes  unequivocally 
yield  results  which  indicate  that  the  sum  of  the 
angles  in  cosmic  triangles  is  either  a  trifle  more 
or  a  trifle  less  than  i8o^ ;  would  we  have  to  conclude 
that  cosmic  space  is  curved,  or  would  we  not  have 
to  look  for  some  concrete  and  specird  cause  for  the 
aherration  of  the  light?  If  the  moon  is  eclijiscd 
while  the  sun  still  appears  on  the  horizon,  it  proves 
only  that  the  refraction  of  the  solar  rays  makes  the 
sun  appear  higher  than  it  really  standi,  if  its  posi- 
tion is  dclcrniincd  hv  a  straight  line-,  hnl  it  dors  not 
refute  the  straight  line  conception  of  geometry. 
?^Ieasurements  of  star-parallaxes  (if  they  could  no 
longer  be  account-cd  for  by  the  personal  equation 


126         THE   FOUNDATIONS  OF   MATHEMATICS. 

of  erroneous  observation),  may  prove  that  ether  can 
sHghtly  deflect  the  rays  of  hght,  but  it  will  never 
prove  that  the  straight  line  of  plane  geometry  is 
really  a  cirve.  We  might  as  well  say  that  the  norms 
of  logic  are  refuted  when  we  make  faulty  observa- 
tions or  whenever  we  are  confronted  by  contradic- 
tory statements.  No  one  feels  called  upon,  on  ac- 
count of  the  many  lies  that  are  told,  to  propose  a 
theory  on  the  probable  curvature  of  logic.  Yet, 
seriously  speaking,  in  the  province  of  pure  being 
the  theory  of  a  curved  logic  has  the  same  right  to 
a  respectful  hearing  as  the  curvature  of  space  in  the 
province  of  the  scope  of  pure  motility. 

Ideal  constructions,  like  the  systems  of  geom- 
etry, logic,  etc.,  cannot  be  refuted  by  facts.  Our 
observation  of  facts  may  call  attention  to  the  log- 
ical mistakes  we  have  made,  but  experience  can- 
not overthrow  logic  itself  or  the  principles  of  think- 
ing. They  bear  their  standard  of  correctness  in 
themselves  which  is  based  upon  the  same  principle 
of  consistency  that  pervades  any  system  of  actual 
or  purely  ideal  operations. 

But  if  space  is  not  round,  are  we  not  driven  to 
the  other  horn  of  the  dilemma  that  space  is  infinite? 

Perhaps  we  are.  What  of  it?  I  see  nothing 
amiss  in  the  idea  of  infinite  space. 

By  the  by,  if  objective  space  were  really  curved, 
would  not  its  twist  be  dominated  in  all  probability 
by  more  than  one  determinant  ?  Why  should  it  be 
a  curvature  in  the  plane  which  makes  every  straight 
line  a  circle?     Might  not  the  plane  in  which  our 


MATHEMATICS  AND  METAGEOMETRY.  12/ 

straightest  line  lies  be  also  possessed  of  a  twist  so 
as  to  give  it  the  shape  of  a  flat  screw,  which  would 
change  every  straightest  line  into  a  spiral?  But 
the  spiral  is  as  infinite  as  the  straight  line.  Ob- 
viously, curved  space  does  not  get  rid  of  infinitude; 
besides  the  infinitely  small,  which  w'ould  not  l^e 
thereby  eliminated,  is  not  less  troublesome  than  the 
infinitely  great. 

THE    TKACHING    OF    MATHEMATICS. 

As  has  been  pointed  out  before,  Euclid  avoided 
the  word  axiom,  and  I  believe  with  Grassmann,  that 
its  omission  in  the  Elements  is  not  accidental  init 
the  result  of  well-considered  intention.  1lie  intro- 
duction of  the  term  among  Euclid's  successors  is 
due  to  a  lack  of  clearness  as  to  the  nature  of  geom- 
etry and  the  conditions  through  which  its  funda- 
mental notions  originate. 

Tt  may  Ije  a  flaw  in  the  luiclidcan  lUcuioifs  that 
the  construction  of  the  ])lane  is  presu])poscd.  Imt  it 
does  not  invalidate  the  details  of  his  glorious  work 
which  will  forever  remain  classical. 

Tlic  in\-ention  of  other  geometries  can  oiilv  serve 
to  illustrate  the  truth  that  all  geometries,  the  plane 
geometry  of  Euclid  included,  are  a  priori  construc- 
tions, and  were  not  for  obvious  reasons  luiclid's 
j)lane  geometry  preferable,  other  systems  migiit  as 
well  be  employed  for  tin-  purpose  of  space-determi- 
nation. Neither  liomaloidality  nor  curvature  be- 
longs to  space;  they  belong  to  the  several  systems  of 


128        THE   FOUNDATIONS   OF   MATHEMATICS. 

manifolds  that  can  be  invented  for  the  determina- 
tion of  the  juxtapositions  of  things,  called  space. 

If  I  had  to  rearrange  the  preliminary  expositions 
of  Euclid,  I  would  state  first  the  Coniuion  Notions 
which  embody  those  general  principles  of  Pure  Rea- 
son and  are  indispensable  for  geometry.  Then  I 
would  propose  the  Postulates  which  set  forth  our 
own  activity  (viz.,  the  faculty  of  construction)  and 
the  conditions  under  which  we  intend  to  carry  out 
our  operations,  viz.,  the  obliteration  of  all  particu- 
larity, characterizable  as  "anyness  of  motion." 
Thirdly,  I  would  describe  the  instruments  to  be 
employed:  the  ruler  and  the  pair  of  compasses; 
the  former  being  the  crease  in  a  plane  folded  upon 
itself,  and  the  latter  to  be  conceived  as  a  straight 
line  (a  stretched  string)  one  end  of  which  is  sta- 
tionary while  the  other  is  movable.  And  finally  I 
would  lay  down  the  Definitions  as  the  most  elemen- 
tary constructions  which  are  to  serve  as  tools  and 
objects  for  experiment  in  the  further  expositions 
of  geometry.  There  w^ould  be  no  mention  of  axioms, 
nor  would  we  have  to  regard  anything  as  an  as- 
sumption or  an  hypothesis. 

Professor  Hilbert  has  methodically  arranged 
the  principles  that  underlie  mathematics,  and  the 
excellency  of  his  work  is  universally  recognized.'' 
It  is  a  pity,  however,  that  he  retains  the  term 
"axiom,"  and  we  would  suggest  replacing  it  by 
some  other  appropriate  word.     "Axiom"  with  Hil- 

°  The  Foundations  of  Geometry,  The  Open  Court  Pub.  Co., 
Chicago,  1902. 


MATHEMATICS   AND   METAGEOMETRY.  I2g 

bert  does  not  mean  an  obvious  truth  that  does  not 
stand  in  need  of  proof,  but  a  principle,  or  rule,  viz., 
a  formula  describing  certain  general  characteristic 
conditions. 

iMathematical  space  is  an  ideal  construction,  and 
as  such  it  is  a  priori.  But  its  apriority  is  not  as 
rigid  as  is  the  apriority  of  logic.  It  presupposes 
not  only  the  rules  of  pure  reason  but  also  our  own 
activity  (viz.,  pure  motility)  both  being  sufficient 
to  create  any  and  all  geometrical  figures  a  priori. 

Boundaries  that  are  congruent  with  themselves 
being  limits  that  are  unique  recommend  themselves 
as  standards  of  measurement.  Hence  the  signifi- 
cance of  the  straight  line,  the  plane,  and  the  right 
angle. 

The  theorem  of  parallels  is  only  a  side  issue  of 
the  implications  of  the  straight  line. 

The  postulate  that  figures  of  the  same  relations 
are  congruent  in  whatever  place  they  may  be,  and 
also  that  figures  can  be  drawn  similar  to  any  figure, 
is  due  to  our  abstraction  which  creates  the  condition 
of  anyness. 

The  teaching  of  mathematics,  now  utterly  neg- 
lected in  the  ])ublic  schools  and  not  specially  favored 
in  the  high  schools,  should  begin  early,  but  Tuiclid's 
method  with  his  ])edantic  proj)osilions  and  i)r()(>fs 
shoulfl  be  replaced  by  construction  wf)rk.  I  x't  chil- 
dren begin  geometry  by  doing,  not  by  reasoning, 
'i'he  reasoning  faculties  are  not  yet  sufTicicntly  de- 
\'eloj)ed  in  a  child.  Abstract  reasoning  is  tedicnis, 
but  if  it  comes  in  as  an  inridt-nlal  aid  to  construe- 


130        THE  FOUNDATIONS  OF  MATHEMATICS. 

tion,  it  will  be  welcome.  Action  is  the  main-spring 
of  life  and  the  child  will  be  interested  so  long  as 
there  is  something  to  achieve.^ 

Lines  must  be  divided,  perpendiculars  dropped, 
parallel  lines  drawn,  angles  measured  and  trans- 
ferred, triangles  constructed,  unknown  quantities 
determined  with  the  help  of  proportion,  the  nature 
of  the  triangle  studied  and  its  internal  relations 
laid  down  and  finally  the  right-angled  triangle  com- 
puted by  the  rules  of  trigonometry,  etc.  All  in- 
struction should  consist  in  giving  tasks  to  be  per- 
foniicd,  not  theorems  to  he  proved;  and  the  pupil 
should  find  out  the  theorems  merely  because  he 
needs  them  for  his  construction. 

In  the  triangle  as  well  as  in  the  circle  we  should 
accustom  ourselves  to  using  the  same  names  for  the 
same  parts. '^ 

Every  triangle  is  ABC.  The  angle  at  A  is  al- 
ways a,  at  B  ^,  at  C  y.  The  side  opposite  A  is  a, 
opposite  B  h,  opposite  C  c.  Altitudes  (heights)  are 
ha)  hbt  he.  The  lines  that  from  A,  B,  and  C  pass 
through  the  center  of  gravity  to  the  middle  of  the 
opposite  sides  I  propose  to  call  gravitals  and  would 
designate  \\\^vciga,gb^gc-  The  perpendiculars  erected 
upon  the  middle  of  the  sides  meeting  in  the  center  of 
the  circumscribed  circle  are /«, /^,  Z^.  The  lines 
that  divide  the  angles  a,  /S,  y  and  meet  in  the  center 

°  Cp.  the  author's  article  "Anticipate  the  School"  {Open  Court. 
1899,  p.  747). 

'  Such  was  the  method  of  my  teacher.  Prof.  Hermann  Grass- 
mann. 


MATHEMATICS  AND  METAGEOMETRY.  I3I 

of  the  inscribed  circle  I  propose  to  call  "dichotoms"^ 
and  would  designate  them  as  d,,,  dt,  d,.  The  radius 
of  the  circumscribed  circle  is  r,  of  the  inscribed 
circle  p,  and  the  radii  of  the  three  ascribed  circles 
are  p^,  pty  pc  The  point  where  the  three  heights 
meet  is  H  ;  where  the  three  gravitals  meet,  G;  where 
the  three  dichotoms  meet,  O."  The  stability  of 
designation  is  very  desirable  and  perhaps  indispen- 
sable for  a  clear  comprehension  of  these  important 
interrelated  parts. 

"  From  5(xoro/ios.  I  purposely  avoid  tlie  term  bisector  and  also 
the  term  median,  the  former  because  its  natural  abbreviation  b  is 
already  appropriated  to  the  side  opposite  to  the  point  B,  and  the 
latter  because  it  has  beeen  used  to  denote  sometimes  the  gravitals 
and  sometimes  the  dichotoms.  It  is  thus  reserved  for  general  use 
in  the  sense  of  any  middle  lines. 

'The  capital  of  the  Greek  p  is  objectionable,  because  it  cannot  be 
distinguished  from  the  Roman  P. 


EPILOGUE. 

WHILE  matter  is  eternal  and  energy  is  in- 
destructible, forms  change;  yet  there  is  a 
feature  in  the  changing  of  forms  of  matter  and 
energy  that  does  not  change.  It  is  the  norm  that 
determines  the  nature  of  all  formations,  commonly 
called  law  or  uniformity. 

The  term  ''norm"  is  preferable  to  the  usual 
word  "law"  because  the  unchanging  uniformities 
of  the  domain  of  natural  existence  that  are  formu- 
lated by  naturalists  into  the  so-called  "laws  of  na- 
ture," have  little  analogy  with  ordinances  properly 
denoted  by  the  term  *'law."  The  "laws  of  nature" 
are  not  acts  of  legislation ;  they  are  no  ukases  of  a 
Czar-God,  nor  are  they  any  decrees  of  Fate  or  of 
any  other  anthropomorphic  supremacy  that  sways 
the  universe.  They  are  simply  the  results  of  a 
given  situation,  the  inevitable  consequents  of  some 
event  that  takes  place  under  definite  conditions. 
They  are  due  to  the  consistency  that  prevails  in 
existence. 

There  is  no  compulsion,  no  tyranny  of  external 
oppression.  They  obtain  by  the  internal  necessity 
of  causation.  What  has  been  done  produces  its 
proper   effect,   good  or  evil,   intended  or  not   in- 


EPILOGUE.  133 

tended,  pursuant  to  a  necessity  which  is  not  dy- 
namical and  from  without,  hut  loj^ical  and  from 
within,  yet,  for  all  that,  none  the  less  inevitable. 
The  basis  of  every  so-called  "law  of  nature"  is 
the  norm  of  formal  relations,  and  if  we  call  it  a  law 
of  form,  we  nuist  bear  in  mind  that  the  term  "law" 
is  used  in  the  sense  of  uniformity. 

Form  (or  rather  our  comprehension  of  the  for- 
mal and  of  all  that  it  implies)  is  the  condition  that 
dominates  our  thinking  and  constitutes  the  norm 
of  all  sciences.  From  the  same  source  we  derive 
the  principle  of  consistency  which  underlies  our 
ideas  of  sameness,  uniformity,  rule,  etc.  This  norm 
is  not  a  concrete  fact  of  existence  but  the  universal 
feature  that  i)ermeates  both  the  anyness  of  our 
mathematical  constructions  and  tlie  anyness  of  ob- 
jective conditions.  Its  application  ])roduces  in  the 
realm  of  mind  the  a  priori,  and  in  the  domain  of 
facts  the  uniformities  of  events  whicli  our  scientists 
reduce  to  formulas,  called  laws  of  nature.  On  a 
superficial  inspection  it  is  pure  nothingness,  Iml  in 
fact  it  is  universality,  eternality,  and  omni]^resence ; 
and  it  is  the  factor  objectively  of  the  world  order 
and  subjectively  of  science,  the  latter  being  man's 
capal)ility  of  reducing  the  innumerable  sense-im- 
pressions of  experience  to  a  methodical  sysU-ni  of 
knowledge. 

Faust,  seeking  the  ideal  of  beauty,  is  ruKised  to 
search  for  it  in  the  domain  of  the  eternal  tyi)es  of 
existence,  which  is  the  omnipresent  Nowhere,  tin- 
ever-enduring  Never.     Mepbistopheles  calls  it  the 


134        THE  FOUNDATIONS  OF  MATHEMATICS. 

Naught.  The  norm  of  being,  the  foundation  of 
natural  law,  the  principle  of  thinking,  is  non-exis- 
tent to  Mephistopheles,  but  in  that  nothing  (viz., 
the  absence  of  any  concrete  materiality,  implying  a 
general  anyness)  from  which  we  weave  the  fabric 
of  the  purely  formal  sciences  is  the  realm  in  which 
Faust  finds  "the  mothers"  in  whom  Goethe  personi- 
fies the  Platonic  ideas.  When  Mephistopheles  calls 
it  "the  nothing,"  Faust  replies: 

"In  deinem  Nichts  hoff'  ich  das  All  zu  finden." 
[Tis  in  thy  Naught  I  hope  to  find  the  All] 

And  here  we  find  it  proper  to  notice  the  analogy 
which  mathematics  bears  to  religion.  In  the  his- 
tory of  mathematics  we  have  first  the  rigid  presen- 
tation of  mathematical  truth  discovered  (as  it  were) 
by  instinct,  by  a  prophetic  divination,  for  practical 
purposes,  in  the  shape  of  a  dogma  as  based  upon 
axioms,  which  is  followed  by  a  period  of  unrest, 
being  the  search  for  a  philosophical  basis,  which 
finally  leads  to  a  higher  standpoint  from  which, 
though  it  acknowledges  the  relativity  of  the  primi- 
tive dogmatism,  consists  in  a  recognition  of  the 
eternal  verities  on  which  are  based  all  our  thinking, 
and  being,  and  yearning. 

The  "Naught"  of  Mephistopheles  may  be  empty, 
but  it  is  the  rock  of  ages,  it  is  the  divinity  of  exist- 
ence, and  we  might  well  replace  "All"  by  "God," 
thus  intensifying  the  meaning  of  Faust's  reply,  and 
say: 

"  Tis  in  thy  naught  T  hope  to  find  my  God." 


EPILOGUE.  135 

The  norm  of  Pure  Reason,  the  factor  that  shapes 
the  world,  the  eternal  Logos,  is  omnipresent  and 
eternal.  It  is  God.  The  laws  of  nature  have  not 
been  fashioned  by  a  creator,  they  are  part  and  parcel 
of  the  creator  himself. 

Plutarch  quotes  Plato  as  saying  that  God  is 
always  geometrizing.^  In  other  words,  the  purely 
formal  theorems  of  mathematics  and  logic  are  the 
thoughts  of  God.  Our  thoughts  are  fleeting,  but 
God's  thoughts  are  eternal  and  omnipresent  veri- 
ties. They  are  intrinsically  necessary,  universal, 
immutable,  and  the  standard  of  truth  and  right. 

Matter  is  eternal  and  energy  is  indestructible, 
but  there  is  nothing  divine  in  either  matter  or  en- 
ergy. That  which  constitutes  the  divinity  of  the 
world  is  the  eternal  principle  of  the  laws  of  ex- 
istence. That  is  the  creator  of  the  cosmos,  the 
norm  of  truth,  and  the  standard  of  right  and  wrong. 
If  incarnated  in  living  beings,  it  produces  mind 
and  it  continues  to  be  the  source  of  inspiration  for 
aspiring  mankind,  a  refuge  of  the  struggling  and 
storm-tossed  sailors  on  the  ocean  of  life,  and  the 
holy  of  holies  of  the  religious  devotee  and  wor- 
shiper. 

The  norms  of  logic  and  of  mathematics  are 
uncrcatc  and  uncreatable,  they  arc  irrefragable  and 
immutable,  and  no  power  on  earth  or  in  heaven  can 
change  them.  We  can  imagine  that  the  world  was 
made  by  a  great  world  builder,  but  we  cannot  think 

ThitarcViiis  Cmwiria.  VTTT.  2:  "-tSi  JW&rwv  fXtyt  t6i'  0»Ar  arl  yrui- 
nfTpfii',  Having  hiintefl  in  vain  ior  ilir  famous  passaRc,  I  am  in- 
debted for  the  reference  to  Professor  Ziwct  of  .Ann  .Xrbor,  Midi. 


136         THE   FOUNDATIONS   OF   MATHEMATICS. 

that  logic  or  arithmetic  or  geometry  was  ever  fash- 
ioned by  either  man,  or  ghost,  or  god.  Here  is  the 
rock  on  which  the  old-fashioned  theology  and  all 
mythological  God  -  conceptions  must  founder.  If 
God  were  a  being  like  man,  if  he  had  created  the 
world  as  an  artificer  makes  a  tool,  or  a  potter  shapes 
a  vessel,  we  would  have  to  confess  that  he  is  a  lim- 
ited being.  He  might  be  infinitely  greater  and  more 
powerful  than  man,  but  he  would,  as  much  as  man, 
be  subject  to  the  same  eternal  laws,  and  he  would, 
as  much  as  human  inventors  and  manufacturers, 
have  to  mind  the  multiplication  tables,  the  theorems 
of  mathematics,  and  the  rules  of  logic. 

Happily  this  conception  of  the  deity  may  fairly 
well  be  regarded  as  antiquated.  We  know  now  that 
God  is  not  a  big  individual,  like  his  creatures,  but 
that  he  is  God,  creator,  law,  and  ultimate  norm  of 
everything.  He  is  not  personal  but  superpersonal. 
The  qualities  that  characterize  God  are  omnipres- 
ence, eternality,  intrinsic  necessity,  etc.,  and  surely 
wherever  we  face  eternal  verities  it  is  a  sign  that 
we  are  in  the  presence  of  God, — not  of  a  mytholog- 
ical God,  but  the  God  of  the  cosmic  order,  the  God 
of  mathematics  and  of  science,  the  God  of  the  human 
soul  and  its  aspirations,  the  God  of  will  guided  by 
ideals,  the  God  of  ethics  and  of  duty.  So  long  as  we 
can  trace  law  in  nature,  as  there  is  a  norm  of  truth 
and  untruth,  and  a  standard  of  right  and  wrong, 
we  need  not  turn  atheists,  even  though  the  tradi- 
tional conception  of  God  is  not  free  from  crudities 
and  mythological  adornments.     It  will  be  by  far 


EPILOGUE.  1.^7 

preferable  to  purity  our  conception  of  God  and  re- 
place the  traditional  notion  which  during  the  un- 
scientific age  of  human  development  served  man  as 
a  useful  surrogate,  by  a  new  conception  of  Ciod, 
that  should  be  higher,  and  nobler,  and  better,  be- 
cause truer. 


INDEX. 


Absolute,  The,  25. 

Anschaiiung,  82,  97. 

Anyness,  46ff.,  76;  Space  founded  on, 
60. 

Apollonius,  31. 

A  posteriori,  43,  60. 

A  priori,  38,  64;  and  the  purely  for- 
mal, 4off;  Apparent  arbitrariness  of 
the,  96ff;  constructions,  112;  con- 
structions. Geometries  are,  127; 
constructions  verified  by  experience, 
122;  Geometry  is,  ii9ff;  is  ideal. 
44;  Source  of  the,  51;  The  logical, 
54;  The  purely,  55;  The  rigidly,  54, 
55- 

Apriority  of  different  degrees,  49fT; 
of  mathematical  space,  121,  129;  of 
space-measurement,  logff;  Problem 
of,  36. 

Archimedes,   31. 

As  if,  79. 

Astral   geometry,    15. 

Atomic  fiction,   81. 

Ausdelinungslehre,  Grassmann's,  28, 
29n.,   30. 

"Axiom,"  Euclid  avoided,  i,  127; 
Hubert's  use  of,    128. 

Axioms,    iff;   not   Common   Notions,  4. 

Ball,    Sir    Robert,    on    the    nature    of 

space,    i23f. 
Bernoulli,   9. 

BcsscI,   Letter  of  Gauss  to,    i2fT. 
Billingslcy,    Sir   H.,   82. 
Molyai,   Janos,   22fT,   98;   translated,  27. 
I'ounflary   concepts,    Utility   of,    74. 
IJoundarics,     78.     129;       produced     by 

halving,    Kvcn,   85,   86. 
Bridges   of   Konigsberg.    io2f. 
Busch,    Wilhclin,    115. 

Carus,  Paul,  Fundamental  Problems, 
39n  ;  Kant's  Prolegomena,  39n,  1 2211 ; 
Primer  of  Philosnf'hv    ,(on. 


Causation,  a  priori,  53 ;  and  trans- 
formation,   54;    Kant   on,   40. 

Cayley,  25. 

Chessboard,  Problem  of,  101. 

Circle,  Squaring  of  the,  104;  the  sim- 
l>Iest  curve.   75. 

Classification,    79. 

Clifford,  16,  32,  60;  Plane  constructed 
by,  69. 

Common   notions,   2,   4,    128. 

Conite,   38. 

Concreteness,  Purely  formal,  absence 
of,    60. 

Continuum,   78ff. 

Curved  space,   106;  Ilelmholtz  on,  113. 

Definitions  of   Euclid,    i,    128. 
Dclboeuf,    B.   J.,   27. 
Dc    Morgan,    Augustus,    10. 
Determinism  in  mathematics,   104. 
Dimension,    Definition   of,    85. 
Dimensions,   Si)ace  of  four,  9ofT. 
Directions  of  space.   Infinite,    117. 
Discrete    units,    78fF. 
Dual  number,  89. 

Edward's  Dream,    115. 

I'.gg-shaped   body,    i  ijf. 

KIIi|)tic   geometry,    25. 

I'.mpiricism,  Transcendentalism  and, 
38fT. 

Kngel,   I'riedrich,  26. 

I'.uclid,  1-4,  3 1  IT;  avoided  "axiom,"  1. 
127;  Ex))osifionsof,  rearranged,  128; 
Malsted  on,   3 if. 

I'.iiclidcan   geoinrtry,   classical,   31,  121. 

I'.ven  boundaries,  122;  as  standards  of 
measurement,  6gfl,  85,  86;  ptixliii-ed 
by   halving,   85,   86. 

I'.xperiencc,  Physiological  si)ace  orig- 
inates  through,    65. 

Faust,   133. 

I-'icliliotiH   spaces.    loofT. 


I40 


THE    FOUNDATIONS   OF   MATHEMATICS. 


Flatland,    115. 

Form,  60,    133;  and  reason,  48. 

Four-dimensional  space  and  tridimen- 
sional beings,  93. 

I'our   dimensions,    109;    Space  of,  gofT. 

Fourth  dimension,  25;  illustrated  by 
mirrors,    93ff. 

Gauss,    I  iff;   his  letter  to  Bessel,    laff; 

his  letter  to  Taurinus,   6f,    i^i. 
Geometrical   construction,    Definiteness 

of,   99ff. 
Geometry,   a  priori,    iigiif;    Astral,    15; 

FUiptic,  25;  Question  in,  -j.  73,  121. 
Geometries,  a  priori  constructions,  127. 
God,   Conception  of,   136. 
Grassmann,    27 ff,    127,    i3on. 

Ilalsted,  Geprge  Bruce,  4n,  2on,  23, 
26,     27,  28n,    10 1 ;  on  Euclid,  3 if. 

Helmholtz,  26,  83;  on  curved  space, 
113;  on  two-dimensional  beings, 
I  I  of. 

Hubert's  use  of  "axiom,"    128. 

Honialoidal,    18,    74. 

Homogeneity   of   space,   66ff. 

Hypatia,    31. 

"Ideal"  and  "subjective,"  Kant's  iden- 
tification  of,   44f;   not  synonyms,  64. 

Infinite  directions  of  space,  117;  di- 
vision of  line,  117;  not  mysterious, 
118;  Space  is,  116,  126;  Time  is, 
116. 

Infinitude,    ii6ff. 

Kfint,  35,  40.  6t,  84;  and  the  a  priori, 
36,  38;  his  identification  of  "ideal" 
and  "subjective,"  44f;  his  term 
Anschauung,  32,  97;  his  use  of 
"transcendental,"  41. 

Kant's  Prolegomena,   jgn. 

Keyser,  Cassius  Jackson,   yj. 

Kineniatoscope,    80. 

Klein,   Felix,  25. 

Konigsberg,   Seven  bridges  of,   i02f. 

Lagrange,    lof. 

Lambert,    Johann    Ileinrich,    gf. 

Laws   of  nature,    132. 

Legendre,    i  i. 

Line  created  by  construction.  83:  in- 
dependent of  position,  62;  Infinite 
division  of,  117;  Shortest,  84: 
Straightest,    75,    127. 


Littre,  38. 

Lobatchevsky,    10,   2off,   75,   98;    trans- 
lated,   27. 
Lobatchevsky's     Theory    of    Parallels, 

lOI. 

Logic  is  static,  53. 

Mach,    Ernst,    2y,   65. 

Mathematical    space,    63ff,    67,    109;    a 

priori,   65;   Apriority  of,    121,    129. 
Mathematics,   Analogy  of,   to   religion, 

134;    Determinism    in,    104;    Reality 

of,    77;   Teaching  of,    i27ff. 
Measurement,      Even     boundaries     as 

standards   of,    69ff,    85,    86;    of    star 

parallaxes,    125;    Standards   for,    74. 
Mental  activity.   First  rule  of,  79. 
Metageometry,     sff;     History    of,     2O; 

Mathematics   and,    82ff. 
Mill,   John   Stuart,   38. 
Mind    develops    through    uniformities, 

52;   Origin  of,   51. 
jNIirrors,    Fourth   dimension   illustrated 

by,   93  ff- 
Monist,    4n,    27,    125. 

Names,  Same,  for  parts  of  figures,  130. 

Nasir   Eddin,   7. 

Nature,     16;    a    continuum,    78;    Laws 

of,    132. 
Newcomb,    Simon,    25. 

Open   Court,   2on,    iisn,    i3on. 
Order   in   life   and   arithmetic,    80. 

Pangeometry,    22. 

Pappus,    31. 

Parallel  lines  in  spherical  space,  84; 
theorem,  4n,   25f,  98,    129. 

Parallels,   Axiom   of,   3. 

Path  of  highest  intensity  a  straight 
line,   58. 

Peirce,  Charles  S.,  on  the  nature  of 
space,   i23f. 

Physiological  space,  63ff,  67;  origi- 
nates   through    experience,    65. 

Plane,  a  zero  of  curvature,  82;  con- 
structed by  Clifford,  69;  created  by 
construction,  83;  Nature  of,  73; 
Significance   of,    129. 

Plato,    135. 

Plutarch,    135. 

Poincare,    H.,   27. 

Point    congruent   with   itself,    71. 


IXDEX. 


141 


Population  of  Great  Britain  deter- 
mined,   iigf. 

Position,   62. 

Postulates,   2,    128. 

Potentiality,  63. 

Proclus,   4,   31. 

Pseudo-spheres,   83. 

Pure  form,  63;  space,  Unif|uencss  of, 
6iff. 

Purely  a  priori.  The,  55:  formal,  ab- 
sence of  concreteness,    60. 

Question   in   geometry,    72,    73.    121. 

Ray  a  final   boundary,   58. 

Reason,   Form  and,  48;   Nature  of,  76. 

Rectangular    pentagon,    98. 

Religion,  Analogy  of  mathematics  to, 
134- 

Riemann,  isff,  37,  62,  96,  104,  io6fT, 
123. 

Right  angle  created  by  construction, 
83;  Nature  of,  73;  Significance  of, 
129. 

Russell,  Bertrand  A.  \V.,  26;  on  non- 
Euclidean    geometry,    1  19. 

Saccheri,    Girolamo,   8f. 

Schlegel,    Victor,    30. 

Schoute,  P.  H.,  3on. 

Schumaker,   Letter  of  Gauss  to,    11. 

Schweikart,    15. 

Sense-experience    and    space,     i22ff. 

Shortest  line,   84. 

Space,  a  manifold,  108;  a  spread  of 
motion,  56ff;  Apriority  of  mathe- 
matical, 121,  129;  curved,  126; 
founded  on  "anyness,"  60;  Helm- 
holtz  on  curved,  113;  Homogeneity 
of,  66ff;  homaloidal,  74;  Infinite  di- 
rections of,  117:  Interference  of, 
loiflF;  is  infinite,  116,  126;  Mathe- 
matical, 63ff,  67,  109;  Mathematical 
and  actual,  62:  of  four  dimensions, 
9ofT;  On  the  nature  of,  i23f;  Phys- 
iological, 63(1,  67;  Sense-experience 
and,  i22ff;  the  juxtaposition  of 
things,  67,  87;  the  possibility  of 
motion,  59;  the  potentiality  of  meas- 
uring, 61;  Uniqueness  of  [)ure,  6ifT. 

Space-conception,  how  far  ii  prIoriT 
S9f:   product  of  pure  activity,   55. 


Space-measurement,  Apriority  of,  logfF; 
X'arious  systems  of,   104!?. 

Spaces,    Fictitious.    io9ff. 

Squaring  of  the  circle,    104. 

Stackel,    Paul,   26. 

Standards  of  measurement.  74;  Even 
boundaries  as,   6gfl,   85,   86. 

.Star  parallaxes.  Measurements  of,  125. 

.Straight  line,  69.  71,  112,  122:  a  path 
of  highest  intensity,  59;  created  by 
construction,  83:  does  not  exist,  72; 
indispensable,  72ft;  Nature  of,  73; 
One  kind  of.  75;  Signiticance  of, 
129;   possible,   74. 

Straightest   line,    75,    127. 

Subjective  and  ideal,  Kant's  identifi- 
cation of,   44f;   not  synonyms,  64. 

Sui)erreal,    The,    76(1. 

Taurinus,   Letter  of  Gauss  to,  6f,   I3f. 

Teaching  of  mathematics,   i27ff. 

Tentamen,   23. 

Theon,  31,  32. 

Theory  of  Parallels,  Lobatchevsky's, 
loi. 

Thought-forms,  systems  of  reference, 
61. 

Three,    The    number,    88f. 

Time  is  infinite,    116. 

"Transcendental,"    Kant's   use   of,    41. 

Transcendentalism  and  Em|)iricism,3s. 
sSflf. 

Transformation,    Causation   and,    54. 

Tridimensional  beings.  Four-dimen- 
sional space  and,  93;  space,  Two- 
dimensional  beings  and,   91. 

Tridimensionality,    84flt. 

Trinity,    Doctrine  of  the,   89. 

Two-dimensional  beings  and  triilimcn- 
sional   space,   91. 

L'niformities,     132;     Mind     develops 

through,  52. 
I'nits,    Discrete.   78ff:    Positing  r)f,   80. 

Wall  is,   John,   7f. 
Whv?    mo. 


Zambcrti,   32. 

Ziwct,   Proffs.sor,    i35n. 


The  Open  Court  Mathematical  Series 


Essays  on  the  Theory  of  Numbers. 

(1)  Continuity  and  Irrational  Numbers,  (2)  The  Nature 
and  Meaning  of  Numbers.  By  Richard  Dedekind.  From 
the  German  by  W.  W.  Beman.  Pages,  115.  Cloth,  7,") 
cents  net.     (3s.  6d.  net.) 

These  essays  mark  one  of  the  distinct  stages  in  the  devel- 
opment of  the  theory  of  numbers.  They  give  the  founda- 
tion upon  which  the  whole  science  of  numbers  may  be  es- 
tablished. The  first  can  be  read  without  any  technical, 
philosophical  or  mathematical  knowledge;  the  second  re- 
quires more  power  of  abstraction  for  its  perusal,  but  power 
of  a  logical  nature  only. 

"A  model  of  clear  and  beautiful  reasoning." 

— Journal  of  Physical  Chemistn/. 

"The  work  of  Dedekind  is  very  fundamental,  and  I  am  glad  to  have  it 
in  this  carefully  wrought  English  version.  I  think  the  book  should  be 
of    much   service   to    American    mathematicians   and    teachers." 

— Prof.  E.  H.  Moore,  University  of  Chicago. 

"Tt  is  to  be  hoped  that  the  translation  will  make  the  essays  better 
known  to  English  mathematicians  ;  they  are  of  the  very  first  importance, 
and  rank  with  the  work  of  Weierstrass,  Kronecker,  and  Cantor  in  the 
same   field." — Nature. 


Klementary  Illiistrations  of  the  Differential 
and  Inte^^ral  Calculus. 

By  AuncsTUs  De  Mok(;.\n.  New  reprint  edition.  With 
subheadings  and  bibliography  of  Knglish  and  foreign  works 
on  the  Calculus.     Price,  cloth,  $1.00  net.     (4s.  Gd  net.) 

"It  alms  not  at  helping  students  to  cram  for  examinations,  but  to  give 
a  scientific  explanation  of  the  rationale  of  these  branches  of  mathe- 
matics. Like  all  that  De  Morgan  wrote,  it  l.s  accurate,  clear  and 
philosophic." — Literary    World,   London. 


On    the    Study    and    Diff  JriiK  ies    of    Mathe- 
matics. 

By  Augustus  De  Morgan.  With  portrait  of  De  Morgan, 
Index,  and  Bibliograi)hies  of  Moclcrn  Works  on  Algebra, 
the  Philosophy  of  Mathematics,  Pangeomctry.  etc.  Pages, 
viii,  288.     Clo'th,  ll.a.'i  net     (5s.  net.) 

"The  point  of  view  Is  unusual  ;  we  are  confronted  by  a  gonlUR,  who. 
like  his  kind,  showH  little  heed  for  cUKlrimary  i  onventlonH.  The  'shiik- 
Ing  up"  which  this  little  work  will  give  to  th<'  young  teacher,  the  nHm- 
Mlus  and  implied  criflclBm.lt  can  furnish  to  the  more  experienced,  make 
its    posseoslnn    most    dfsirnhle." — Mithiiinn    Aliimtuin. 


The  Open  Court  Mathematical  Series 


The  Foundations  of  Geometry. 

By  David  Hilbert,  Ph.  D.,  Professor  of  Mathematics  in 
the  University  of  Gottingen.  With  many  new  additions 
still  unpublished  in  German.  Translated  by  E.  J.  Town- 
send,  Ph.  D,,  Associate  Professor  of  Mathematics  in  the 
Universitv  of  Illinois.  Pages,  viii,  132.  C'oth,  $1.00  net. 
(4s.  6d  net.) 

"Professor  Hilbert  has  become  so  well  known  to  the  mathematical 
world  by  his  writings  that  the  treatment  of  any  topic  by  him  commands 
the  attention  of  mathematicians  everywhere.  The  teachers  of  elemen- 
tary geometry  in  this  country  are  to  be  congratulated  that  it  is  possible 
for  them  to  obtain  in  English  such  an  important  discussion  of  these 
points   by    such   an    authority."— Journal   of  Pedagogy. 

Euclid's  Parallel  Postulate :  Its  Nature, Val- 
idity and  Place  in  Geometrical  Systems. 

By  John  William  Withers,  Ph.  D.  Pages  vii,  193.  Cloth, 
net  $1.25.      (4s.   6d.  net.) 

"This  is  a  philosophical  thesis,  by  a  writer  who  is  really  familiar  with 
the  subject  on  non-Euclidean  geometry,  and  as  such  it  is  well  worth 
reading.  The  first  three  chapters  are  historical  ;  the  remaining  three 
deal  with  the  psychological  and  metaphysical  aspects  of  the  problem  ; 
finally  there  is  a  bibliography  of  fifteen  phges.  Mr.  Withers's  critique, 
on  the  whole,  is  quite  sound,  although  there  are  a  few  passages  either 
vague  or  disputable.  Mr.  Withers's  main  contention  is  that  Euclid's 
parallel  postulate  is  empirical,  and  this  may  be  admitted  in  the  sense 
that  his  argument  requires  ;  at  any  rate,  he  shows  the  absurdity  of 
some  statements  of  the  a  priori  school." — Nature. 

Mathematical  Essays  and  Recreations. 

By   Hermann    Schubert,    Professor   of   Mathematics    in 

Hamburg.     Contents :     Notion  and  Definition  of  Number ; 

Monism   in   Arithmetic;    On   the   Nature   of   Mathematical 

Knowledge ;  The  Magic  Square ;  The  Fourth  Dimension ; 

The  Squaring  of  the  Circle.     From  the  German  by  T.  J. 

McCormack.     Pages,  149.     Cuts,  37.     Cloth,  75  cents  net. 

(3s.  6d.  net.) 
"Professor  Schubert's  essays  make  delightful  as  well  as  instructive 
reading.  They  deal,  not  with  the  dry  side  of  mathematics,  but  with  the 
philosophical  side  of  that  science  on  the  one  hand  and  its  romantic  and 
my.stical  side  on  the  other.  No  great  amount  of  mathematical  knowl- 
edge is  necessary  in  order  to  thoroughly  appreciate  and  enjoy  them. 
They  are  admirably  lucid  and  simple  and  answer  questions  in  which 
every  intelligent  man  is  interested." — Chicar/o  Exiening  Post. 
"They  should  delight  the  jaded  teacher  of  elementary  arithmetic,  who 
l.s  too  liable  to  drop  into  a  mere  rule  of  thumb  system  and  forget  the 
scientific  side  of  his  work.  Their  chief  merit  is  however  their  intel- 
ligibility. Even  the  lay  mind  can  understand  and  take  a  deep  interest 
in  what  the  German  professor  has  to  say  on  the  history  of  magic 
squares,   the   fourth  dimension  and  squaring  of  the  circle." 

— Saturday  Review. 


The  Open  Court  Mathematical  Series 


A  Brief  History  of  Mathematics. 

By  the  late  Dr.  Karl  Fink,  Tubingen,  Germany.  Trans- 
lated by  Wooster  Woodruff  Beman.  Professor  of  Math- 
ematics in  the  University  of  Michigan,  and  David  Eugene 
Smith,  Professor  of  Mathematics  in  Teachers'  College. 
Columbia  University,  New  York  City.  With  biographical 
notes  and  full  index.  Second  revised  edition.  Pages, 
xii,  333.     Cloth,  $1.50  net.     (5s.  6d.  net.) 

"Dr.  Fink's  work  is  the  most  systematic  attempt  yet  made  to  present  a 
compendious    history    of    mathematics." — The    Outlook. 
"This  book   is  the  best  that  has  appeared   in   English.      It  should  find   a 
place   in    the   library   of   every    teacher   of   mathematics." 

— The  Inland  Educator. 

I^ectures  on  Elementary  Mathematics. 

By  Joseph  Louis  L.vgr.xkge.  With  portrait  and  biography 
of  Lagrange.  Translated  from  the  French  by  T.  J.  Mc- 
Cormack.     Pages,  172.     Cloth,  $1.00  net.     (4s.  6d.  net.) 

"Historical  and  methodological  remarks  abound,  and  are  so  woven  to- 
gether with  the  mathematical  material  proper,  and  the  whole  is  so 
vivified  by  the  clear  and  almost  chatty  style  of  the  author  as  to  give 
the  lectures  a  charm  for  the  readers  not  often  to  be  found  in  mathe- 
matical   works." — Bulletin    American    Mathematical    Society. 

A  Scrapbook  of  Elementary  ]\Ia< hematics. 

P.y  W.M.  F.  White,  State  Xormal  School,  New  Paltz,  N. 
Y.  Cloth.  Pages,  24H.  $1.00  net.  (5s.  net.) 
A  collection  of  Accounts,  Essays,  Recreations  and  Notes, 
selected  for  their  conspicuous  interest  from  the  domain  of 
mathematics,  and  calcuiatefl  to  reveal  that  domain  as  a 
world  in  which  invention  and  imagination  arc  prodigiously 
enabled,  and  in  which  the  practice  of  generalization  is  car- 
ried to  extents  tmdreamed  of  by  the  ordinary  thinker,  who 
has  at  his  command  only  the  resources  of  ordinary  lan- 
guage. A  few  of  the  seventy  sections  of  this  attractive 
book  have  the  following  suggestive  titles:  l'"amiliar  Tricks. 
Algebraic  Fallacies.  Geometric  Puzzles.  Linkages.  A  Few 
Surprising  Facts,  Labyrinths.  The  Nature  of  Mathematic.il 
Reasoning,  Alice  in  the  Wonfjerland  of  Mathematics,  ilie 
book  is  supplicfl  witli  l'.iblif)grai)hic  Notes,  Bibliogr.iphii- 
Indcx  and  a  copious  General  Index. 

The  book  Is  Interesting,  valuable  and  sukrcrIIvp.  It  Ir  n  bonk  tli.n) 
really  AIIb  a  long-felt  want.  It  Ih  a  book  that  ahniild  be  In  the  llbrar.v 
of  every  high  scbool  and  on   the  dcKk  of  every  tcTctinr  of  niatliiTnailis.'' 

— Thi-    Eduraitir-Jiturniil. 


The  Open  Court  Mathematical  Series 


Geometric  Exercises  in  Paper-Folding. 

By  T.  SuNDARA  Row.  Edited  and  revised  by  W.  W.  Be- 
MAN  and  D.  E.  Smith.  With  half-tone  engravings  from 
photographs  of  actual  exercises,  and  a  package  of  papers 
for  folding.  Pages,  x,  148.  Price,  cloth,  $1.00  net.  (4s. 
6d.  net.) 

"The  book  is  simply  a  revelation  in  paper  folding.  All  sorts  of  things 
are  done  with  the  paper  squares,  and  a  large  number  of  geometric 
figures  are  constructed   and  explained   in   the  simplest  way." 

— Teachers'  Itistitute. 

Magic  Squares  and  Cubes. 

By  W.  S.  Andrews.  With  chapters  by  Paul  Carus.  L.  S. 
Frierson  and  C.  A.  Browne,  Jr.,  and  Introduction  by 
Paul  Carus.  Price,  $1.50  net.  (7s.6d.net.) 
The  first  two  chapters  consist  of  a  general  discussion  of  the 
general  qualities  and  characteristics  of  odd  and  even  magic 
squares  and  cubes,  and  notes  on  their  construction.  The 
third  describes  the  squares  of  Benjamin  Franklin  and  their 
characteristics,  while  Dr.  Carus  adds  a  further  analysis 
of  these  squares.  The  fourth  chapter  contains  "Reflections 
on  Magic  Squares"  by  Dr.  Carus,  in  which  he  brings  out 
the  intrinsic  harmony  and  symmetry  which  exists  in  the 
laws  governing  the  construction  of  these  apparently  mag- 
ical groups  of  numbers.  Mr.  Frierson's  "Mathematical 
Study  of  Magic  Squares,"  which  forms  the  fifth  chapter, 
states  the  laws  in  algebraic  formulas.  Mr.  Browne  con- 
tributes a  chapter  on  "Magic  Squares  and  Pythagorean 
Numbers."  in  wh'ch  he  shows  the  importance  laid  by  the 
ancients  on  strange  and  mystical  combinations  of  figures. 
The  book  closes  with  three  chapters  of  generalizations  in 
which  Mr.  Andrews  discusses  "Some  Curious  Magic 
Squares  and  Combinations,"  "Notes  on  Various  Con- 
structive Plans  by  Which  Magic  Squares  May  Be  Classi- 
fied," and  "The  Mathematical  Value  of  Magic  Squares." 

"The  examples  are  numerous :  the  laws  and  rules,  some  of  them 
original,  for  making  squares  are  well  worked  out.  The  volume  is 
attractive  in  appearance,  and  what  is  of  the  greatest  importance  in 
such    a    work,    the    proof-reading    has    been    careful." — The    Nation. 

The  Foundations  of  Mathematics. 

A  Contribution  to  The  Pliilosophy  of  Geometry.  Bv  Dr. 
Paul  Cakus.  140  pages.  Cloth.  Gilt  top.  7,5  cents  net. 
(:?s.  fid.   net.) 


The  Open  Court  Publishing  Co. 

37S-3SS  Wabash  Avenue  Chicago 


U  00° 


498  375    i 


